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พฤติกรรมแบบเกาส์ของเคอร์เนลความร้อน Gaussian
พฤติกรรมแบบเกาส์ของเคอร์เนลความร้อน Gaussian Behavior of Heat Kernels สันติ ทาเสนา* ภาควิชาคณิตศาสตร์ คณะวิทยาศาสตร์ มหาวิทยาลัยเชียงใหม่ Santi Tasena* Department of Mathematics, Faculty of Science Chiang Mai University บทคัดย่อ สมการความร้อนคือสมการที่ใช้อธิบายการกระจายความร้อนบนปริภูมิหนึ่งๆ ณ เวลาใดๆ โดยทั่วไปค�ำตอบของสมการความร้อน จะอยู่ในรูปปริพันธ์ของผลคูณระหว่างฟังก์ชันความร้อนเริ่มต้นและเคอร์เนลความร้อน ดังนั้นพฤติกรรมของเคอร์เนลความร้อนจึง มีผลกระทบโดยตรงต่อพฤติกรรมของค�ำตอบของสมการความร้อน บทความนี้จะยกตัวอย่างความสัมพันธ์ข้างต้น ซึ่งก็คือความสัมพันธ์ ระหว่างพฤติกรรมแบบเกาส์ของเคอร์เนลความร้อนและอสมการฮาร์แนค รวมถึงผลกระทบที่พฤติกรรมทั้งสองมีต่อสมบัติทางเรขาคณิต ได้แก่สมบัติทวีคูณและอสมการปวงกาเรของปริภูมินั้นด้วย โดยผู้เขียนหวังว่าบทความนี้จะเป็นความรู้เบี้องต้นให้กับผู้ที่สนใจท�ำวิจัย ในสาขานี้ต่อไป ค�ำส�ำคัญ :อสมการฮาร์แนค การประมาณค่าเคอร์เนลความร้อน สมบัติทวีคูณ อสมการปวงกาเร Abstract Heat equations are used to explain the distribution of heat over spaces and time. Typically, solutions of heat equations can be written as integrations of their initial heat profiles and heat kernels. Thus, heat kernel behavior directly influences the behavior of solutions of heat equations and vice versa. This article will review one such relation: the Gaussian behavior of heat kernel and the (parabolic) Harnack inequality. This article will also discuss how these two properties relate to the geometric properties such as doubling property and Poincare inequality of the underlying spaces. Hopefully, this will serve as an introduction to those interested in working in this field. Keywords : Harnack inequality, heat kernel estimates, doubling property, Poincare inequality -------------------------------------------------- *E-mail: santi.tasena@cmu.ac.th สันติ ทาเสนา / วารสารวิทยาศาสตร์บูรพา. 19 (2557) 1 : 169-177 169 11 One question researcher might thatkernel whether or not the 11 (Nash, Oneare question anyreplacing researcher might ask any is1967), that whether or operators. notask the isheat associated to 10 than sixty General heat Nash equations given Moser by the Laplacian with other Less years ago, 1958), (Moser, 1961, 1964, and heat Aronson the 12 heat operator exhibits the same behavior as that of the heat kern 12 heat operator exhibits the same behavior as that of the heat kernel associated to Lapla 11 One 1967, question any1971) researcher might askworked is that on whether or not the heatcan kernel associated provided that the initial hea (Aronson, 1968, independently this problem which be deduced that7to that 13of Less than sixty years ago, Nash (Nash, 1958), Moser (Moser, 19 13 Less than sixty years ago, Nash (Nash, 1958), Moser (Moser, 1961, 1964, 1967), and Aro 12 heat operator exhibits the same behavior as that the heat kernel associated to the Laplacian. 7 provided that the initial heat profile satisfies some regularity assumptions. The 7 provided that the initial heat profile satisfies some regularity assumptions. The the result holds for uniformly elliptic operators on that i.e.theforinitial heatheat operators of the form satisfies provided profile 7 some regularity ass 8 function 14 1971) (Aronson, 1967,1964, 1968, 1971) independently worked probl 14 Nash (Aronson, 1968, independently worked onand thisAronson problem which canonbethis deduced 13 Less than sixty (Nash,1967, 1958), (Moser, 1961, 1967), 2 Moser 6 such years that ago, isis smooth forthe allresult and there exists 2 8 function called the heat kernel associated to the Laplacian 8 function is called the heat kernel associated to the Laplacian 15 holds for uniformly elliptic operators on i.e. 15 the result holds for uniformly elliptic operators on i.e. for heat operators of thefo 8 function called to 14 (Aronson, 1967, 1968, 1971) independently worked on this problem which can beisdeduced thatheat kernel associated 9 the . Introduction Dirichlet Forms 1 Introduction 7 for which for all 15 99 the result holds ..for uniformly elliptic operators on i.e. for heat operators of the form 16 such that is smooth for all 2 16 such that . is smooth for10 all 9 Generaland heatthere equatie thenThe classical heat equation is given by To simplify the analysis, the author assumes 8 . Since the result has been progressed in many directions. This article will focus on Introduction 2 The classical heat equation is given by 1610 such that is smooth for all and there exists 11 One any research General are given replacing the Laplacian other 10 General heat heat equations equations are Introduction given bywhich replacing the heat Laplacian with other heatbyoperators. operators. 17General for with which 17 forby for all equations are given the Laplacian with other 2 heat 2question 9 results related to the perturbation of the heat10equation using Dirichlet forms approach. that the topological space X isreplacing locally compact but 12 heat operator exhibits theasos 11 One question any researcher might ask is that whether or not the heat kernel associated to that 11 One question any researcher might ask is that whether or not the heat kernel associated to that , where the function 0 18 . Since then the result has been progressed in many directions given by , where the function 3 satisfying the above equation is called a solutio 18 . Since then the result has been progressed in many directions. This article will focus 11 One question any researcher might ask is that whether or not the heat kernel 17 The classical for which for all Introduction heat equation is given by , where the function2 0 non-compact, and Hausdorff, the associated 13 Less than sixty ago,to 12 the behavior that of heat kernel to the Laplacian. 12 heat heat operator operator exhibits exhibits19 the same same behavior asto that of the theTypically, heat kernel associated tocountable theusing Laplacian. 19 results related tosecond the perturbation of the heat equation using Dirich equation. is interpreted as the temperature atyears point results related the perturbation of theassociated heat equation Dirichlet forms approach. 124 as heat exhibits same behavior as that of the heat kernel 18 . Since then the result has been progressed inoperator many directions. This article will focus on the Dirichlet Forms e1 heat equation is called a satisfying solution ofthe the classical heat above equation is called a solution of the classical heat satisfying the above equation is called a equation is given by , where the function 14 (Aronson, 1967, 1968, 197 13 Less than sixty years ago, Nash (Nash, 1958), Moser (Moser, 1961, 1964, 1967), and Aronson 13 Less than sixty years ago, Nash (Nash, 1958), Moser (Moser, 1961, 1964, 1967), and Aronson Introduction 1 19 Introduction 20 solution Borel measure ν space onthe X always has full support, and1961, the 20 135heat Less than sixty years ago, Nash (Nash, 1958), Moser (Moser, i.e. the represents evolution of temperature over1964, space1967 and results related to perturbation oftime the equation using Dirichlet forms approach. 2asequation. To simplify thethe analysis, the author assumes that the topological is locally the14 temperature at point and at Typically, is interpreted as the temperature at point and at time (Aronson, 1967, 1968, 1971) independently worked on this problem which can be deduced that 14 (Aronson, 1967, 1968, 1971) independently worked on this problem which can be deduced that 15 the result holds for unifo u(t,x) 2 solution of the classical heat equation. Typically, 21 Dirichlet Forms 21 Dirichlet Forms 14 (Aronson, 1967, 1968, 1971) independently worked on this problem which can b the above equation is called awell-known solution of solution the classical heat 6 and that of the classical 20 form (E,D(E)) onequation L always (X, )satisfies is regular, strongly 3classical compact but non-compact, second Hausdorff, the Borel measure on Introduction heat equation isand given by , Dirichlet where the function 2temperature The classical heat equation is given by ,heat where theνform function 1countable Introduction nsatisfying ofi.e. over space time. It iselliptic 15 the result holds for uniformly operators on i.e. for heat operators of the 15 solution the result holds for uniformly elliptic operators on i.e. for heat operators of the form the represents the evolution of temperature over space and time. It is well-known 22 To simplify the analysis, the author assumes that the 16theontopological 22 To analysis, the author assumes that isthatlo 15 simplify the result holds for uniformly elliptic operators i.e. for space heatsuch operator and issupport, interpreted asthe the temperature point interpreted as temperature atatpoint and atlocal, time 21 isfull Dirichlet Forms 4 satisfies has andthe Dirichlet form onthe isand regular, strongly local, and admits the carre du champ operator. on that solution of the classical heat equation satisfies al satisfying above is called a solution of the classical heat The classical heat equation given by , where the function 3 satisfying the above equation is called a solution of classical heat 23 compact but non-compact, second countable and Hausdorff, 16 such that is smooth for all and there exists 16 such that is smooth for all and there exists 23 compact but non-compact, second countable and Hausdorff, the Borel measure on 2 The classical heat equation is given by , where 22 To simplify the analysis, the author assumes that the topological space is locally presents the evolution temperature over space time. It is well-known 16 and such that is17smooth for all for which the a 5 admits the carret >duof champ operator. at time 0 i.e. the solution u represents the evolution Definition 1. A Dirichlet form is a pair (E,D(E)) where ypically, isTypically, interpreted as24the temperature at and at timeofand 24 has full theclassical Dirichlet formis regular, strongly on 4 23 equation. issecond interpreted as point the temperature atsupport, point and at time has fullcountable support, and the Dirichlet form onon local, local 6classical compact but non-compact, Hausdorff, the Borel measure always heat equation satisfies satisfying the above equation is called a solution the heat 3time. It is17well-known satisfying the equation a solution ofregul theh carre 17 for which for all 17 of temperature for over which for allabove 18is called . Since thenfor theall result space and that which provided that the initial heat profile 7carre satisfies some 25 admits du champ tion over space andfor time. Itthe isspace well-known 25 admits du champ operator. 5 Definition i.e.has the solution represents the of temperature over and time. is2 well-known isis aoperator. closed, positive 7 equation. 1.the Aevolution Dirichlet form is a evolution pairthe where aIttime closed, 24represents full support, and of thetemperature Dirichlet form on isinterpreted regular, strongly local, and symmetric, Typically, is interpreted as the temperature at point and at 4 equation. Typically, is as the temperature at point and 19many results relatedThis to the perturb 26 18 .. Since then the result has been progressed many directions. This article will on the 18 classical Since then the result hasheat been progressed inSince manythen directions. This article will focus focus in on the 26 18 satisfies .in the result has been progressed directions. article w solution usome of the classical heat equation 2 form n8 i.e. the heat equation satisfies 6ofpositive that solution of the classical equation satisfies symmetric, densely defined, bilinear on with the following property: for 25 admits the carre du champ operator. satisfies regularity assumptions. The densely defined, bilinear form on L (X, ν ) with the the solution represents the evolution of temperature over space and time. It is well-known 8heat function is the heat and kernel asso satisfies some regularity assumptions. The provided that therelated initialto profileDefinition 5 of i.e. the solution represents the evolution ofheat temperature over space time. It 20 19 the the equation using Dirichlet approach. 19 results results related toheat the perturbation perturbation of the heat equation using Dirichlet forms approach. 27related Definition 1.forms A Dirichlet form is acalled pair where 27 1. A Dirichlet form is a pair where is a cl 26 19 results to the perturbation of the equation using Dirichlet forms appro 9 that anysolution of , the function and . for2 any f ∈ D(E), the function the classical heat equation satisfies following property: 20 20 21 6 that solution of the classical heat equation satisfies densely defined, bilinear on property form 2w 28 positive densely defined, bilinear on Laplacian with theform following 20regularity 27 profile Definition 1. Aassociated Dirichlet form is is asymmetric, pair where is a closed, is heat called the heat kernel to some the Laplacian ial satisfies assumptions. The Introduction 9with . symmetric, called the28 heatpositive kernel associated to the 0 function equipped the inner product is a 1 any Introduction 22 To simplify the an 21 Dirichlet Forms 21 The set Dirichlet Forms 29 any , the function and and 29 , the function and . Dirichlet Forms 28 positive symmetric, densely defined, 21 bilinear form on with the following property: for equations 10 General heat are given by replacing the Laplacian wi 1 the space, called a Dirichlet space, and its norm associated to its inner product is referred to 22 To simplify the analysis, the author assumes that the topological space is locally 22provided To simplify analysis, the author assumes that the topological space is locally 23 compact but non-compact, The classical heat equation is given by , where the function is called the heat kernel associated to the Laplacian provided that the initial heat profile at initial heat profile satisfies some regularity assumptions. The . that the initial heat profile 7 Hilbert satisfies some regularity assumptions. The 22 the analysis, the.the author assumes thatwith the 2the topological spa 29 any , the function and The classical heat equation iswith given by , where th any The set measure equipped with 30To simplify The produ 30 2 The inner product Introduction 1 non-compact, Introduction 11 setcompact One researcher might ask is that whether orthe notinner the heat k 2by replacing as23 the Dirichlet norm. compact but second countable and Hausdorff, the Borel on always 23 compact but non-compact, second countable andquestion Hausdorff, theequipped Borelsecond measure on always 24 has full the support, and the D 2 2 23 but non-compact, countable and Hausdorff, Borel measure the Laplacian with other heat operators. satisfies some regularity assumptions. The function satisfying theis31 above equation iskernel called aDirichlet solution of the classical heat General heat equations are given by replacing the Laplacian with other heat operators. that the initial heat profile satisfies some regularity assumptions. The 31 Hilbert space, called a Dirichlet space, and its norm associated to Hilbert space, called a space, and its norm associated to its inner product is referr 3 provided inner product is a Hilbert space, The set equipped with the inner product is a provided that the initial heat profile 7 satisfies some regularity assu called the heat associated to the Laplacian 2 8 3024 function is called the heat kernel associated to the Laplacian 3 satisfying the above equation is called a solution of the cla 12 heat operator exhibits the same behavior as that of the heat kernel asso 25 admits the carre du champ has the form on is regular, strongly local, 24 heat hasorfull full support, andkernel the Dirichlet Dirichlet form on, where isthe regular, strongly local, and andon classical equation isand given by function 24 to has full support, and the Dirichlet form is regular, stroo stion. whether notsupport, heat associated that 2theis Theas classical heat equation isthe given by ,towhere the function Typically, interpreted as the temperature at point and at time 32 as the Dirichlet norm. One question any researcher might ask is that whether or not heat kernel associated to that 4 that The regularity assumption means the intersection of and , the set of real32 the Dirichlet norm. 26 31 Hilbert space, called a Dirichlet space, and its norm associated to its inner product is referred 4the equation. Typically, is astothe temperature at point and 19 at 25 admits the carre du operator. 25 are admits the carre du champ champ operator. called ainterpreted Dirichlet space, andcalled its 1958), norm to itsassociated 13called Less than sixty years ago, Nash (Nash, Moser (Moser, 1961, equations given by Laplacian with other heat operators. is25 the heat function is called the heat kernel associated the Laplacian admits the carre duthe champ operator. 33 .valued 33 9that .replacing of1satisfying the heat kernel associated to the Laplacian. 8of function is the associated heat kernel to Introduction he solution represents the evolution temperature space and time. It isunder well-known operator exhibits the same behavior as that of the heat kernel associated to the Laplacian. 5asheat continuous functions with compact support on , is dense in the uniform the above equation is called a over solution of classical heat 26 27 Definition 1. problem A Dirichlet 3226 as the Dirichlet norm. Introduction Introduction 3that satisfying the above equation is called a solution of the classical heat 5 i.e. the solution represents the evolution of temperature over space and time. It is w 26 14 (Aronson, 1967, 1968, 1971) independently worked on this wh searcher might ask is whether or not the heat kernel associated to that 34 The regularity assumption means the intersection of inner product is referred to as the Dirichlet norm. 34 The regularity assumption means the intersection of and , the set of 1958), (Moser, 1961, 1964, 1967), and Aronson 33Moser solution ofkernel the classical heat equation satisfies 6 10 metric. This is necessary. The domain of a Dirichlet form must contain enough continuous Less than sixty years ago, Nash (Nash, 1958), Moser (Moser, 1961, 1964, 1967), and Aronson 27 Definition 1. A Dirichlet form is a pair where is a closed, 27 Definition 1. A Dirichlet form is a pair where is a closed, ypically, is interpreted as the temperature at point and at time eral heat equations are given by replacing the Laplacian with other heat operators. . General heat equations are given by replacing the Laplacian with other heat operators. 28 positive symmetric, densely 2 The classical heat equation is given by , where the function 4 equation. Typically, is interpreted as the temperature at point and at time associated to the Laplacian 9 . 27 Definition 1. A Dirichlet form is satisfies aelliptic pair where 6can that solution the heat equation 15associated the result holds for uniformly operators on stlythe same behavior asregularity that which of the heat kernel to classical the 35 valued continuous functions with compact support i.e. on for , the is heat dens quation isis given by , deduced the function equation given by , where where theof function 35 valued continuous functions with compact support on ,means isof dense in under uni worked on this problem be that 34 The assumption means the intersection of and ,deduced the set realorLaplacian. The regularity assumption the intersection 7 11 functions, otherwise it1971) will not reflect analytic properties ofkernel the underlying spaces. See (Oshima (Aronson, 1967, 1968, independently worked on this problem which can beover that on any researcher might ask is that whether or not the heat associated to that ution represents the evolution of temperature over space and time. It is well-known 28 positive symmetric, densely defined, bilinear form on with the following property: for 28 positive symmetric, densely defined, bilinear form on with the following property: for 29 any , the function One question any researcher might ask is that whether not the heat kernel associated to that 5 i.e. the solution represents the evolution of temperature space and time. It is well-known the function 28 positive symmetric, densely defined, bilinear form on with the followin ago, Nash (Nash, 1958), Moser (Moser, 1961, 1964, 1967), and Aronson 36 metric. This is necessary. The domain of a Dirichlet form mu General heat equations are given by replacing the Laplacian with other heat operators. 36 metric. This is necessary. The domain of a Dirichlet form must contain enough contin 3 satisfying the above equation is called a solution of the classical heat 10 General heat equations are given by replacing the Laplacian with other 16 such that is smooth for all 35 valued functions with support on , D(E) istodense in under the uniform continuous on i.e. for heat of form 8 the al, 2010) forcontinuous further treatment of of regularity ofby Dirichlet forms. gng above equation is aoperators solution ofcompact the classical the above equation isequation called athat solution of the classical heat set of the. un the result holds for uniformly elliptic operators on i.e. for heat operators oftonot the form any General equations are given replacing of and Cassociated set of real-valued 29 ,called the function and ..the 29 any , heat the function and or exhibits the same behavior as the heat kernel associated the Laplacian. noperators ofet the classical heat satisfies 12 heat operator exhibits the same behavior as that ofheat the heat kernel the Laplacian. c(X),it 6 that solution of the classical heat equation satisfies 29 any , the function and 30 The 8, 1971) independently worked on this problem which can be deduced that 37 functions, otherwise will reflect analytic properties One question any researcher might ask is that whether or not the heat kernel associated to that 37 functions, otherwise it will not reflect analytic properties of the underlying spaces. See (Os 11 question might ask isenough that whether not the heat kernel ass equation. Typically, isOne interpreted as theresearcher temperature at point and at or time 36 4heat metric. This is necessary. The domain of atime Dirichlet form must contain continuous classical Aas Dirichlet form is(Nash, strongly local ifany for any with as the temperature at and time is9interpreted interpreted the temperature atpoint point and is30 smooth for all and there exists ixty years ago, Nash (Nash, 1958), Moser (Moser, 1961, 1964, 1967), and Aronson 13 Less than sixty years ago, Nash 1958), Moser (Moser, 1961, 1964, 1967), and Aronson 17 for which the Laplacian with other heat operators. One question functions with compact support on Xthe ,is isaawell-known dense inof CDirichlet such that isatat smooth for all and there exists The set equipped with the inner product is 30operator The set equipped with the inner product is 38 et al, 2010) for further treatment of regularity forms. c(X) uniformly elliptic operators on i.e. for heat operators of the form 31 Hilbert space, called a for Diri 38 et al, 2010) for further treatment of regularity of Dirichlet forms. heat exhibits the same behavior as that of the heat kernel associated to the Laplacian. satisfies some regularity assumptions. The ided that the initial heat profile 37 functions, otherwise it will not reflect analytic properties of the underlying spaces. See (Oshima 30 The set equipped with the inner product 12 heat operator exhibits the same behavior as that of heat kernel associated to 5 i.e. the solution represents the evolution of temperature over space and time. It at time 0967, for some constant .time. The is strong locality is equivalent to Leibnitz rule he evolution ofoftemperature over space time. Itfact isiswell-known the evolution temperature over space and It well-known 7 and satisfies some regularity assump provided that the initial heat profile 1968, 1971) independently worked on this problem which can be deduced that 14 (Aronson, 1967, 1968, 1971) independently worked on this problem which can be deduced that 39 A Dirichlet form is strongly local if 18 . Since then the result has been progressed in many directions. This 32 as the Dirichlet norm. 31 Hilbert space, called a Dirichlet space, and its norm associated to its inner product is referred to 31 Hilbert space, called a Dirichlet space, and its norm associated to its inner product is referred to 39 A Dirichlet form is strongly local if for any any researcher might ask is that whether or not the under the uniform metric. This is necessary. The domain Less than sixty years ago, Nash (Nash, 1958), Moser (Moser, 1961, 1964, 1967), and Aronson 38 et al, 2010) for further treatment of regularity of Dirichlet forms. for all 13 sixtyspace, yearsthere ago, exists Nash (Nash, 1958), (Moser, 1961, 1967 6 for that solution of the classical heat equation satisfies 31 than Hilbert called afor Dirichlet space, and Moser its norm associated to its1964, inner produ hholds that is smooth for allLess and slheat well-known equation satisfies which all heat equation satisfies for elliptic operators on i.e. forsome heat operators of the form 33 tion isfunction called heat kernel associated the Laplacian 15 result holds for uniformly elliptic operators on i.e. heat operators oflocality the 40 for some constant .form The fact isDirichlet strong localit 19the results related to thefor perturbation ofdeduced the the heat equation using form 32 as the Dirichlet norm. 32theuniformly as thekernel Dirichlet norm. 40independently for constant .to The fact is strong iscontinuous equivalent to Leibnitz (Aronson, 1967, 1968, 1971) worked on this problem which can be that 8 is called heat kernel associated to the 39 A Dirichlet form is strongly local if for any with 14 (Aronson, 1967, 1968, 1971) independently worked on this problem which can b 32 as the Dirichlet norm. heat associated to that heat operator exhibits of a Dirichlet form must contain enough at the initial heat profile satisfies some regularity assumptions. The ssed in many directions. This article will focus on the provided that the initial heat profile 7result satisfies someonregularity assumptions. The .such Since then the has been progressed in many directions. This article will focus the 34 The regularity assu 33 33 for all 33 20 that is smooth for all and there exists 16 such that is smooth for all and there exists the result holds for uniformly elliptic operators on i.e. for heat operators of the form 40 for some constant . The fact is strong locality is equivalent to Leibnitz rule . the result holds for uniformly elliptic on i.e. for heat operator eat results equation using forms approach. the same behavior as is that ofof15 the heat kernel associated functions, otherwise it means will notset reflect analytic properties toDirichlet the perturbation thethe heat equation Dirichlet forms approach. 34 The regularity assumption means the intersection of and ,,operators the of real34 related The regularity assumption means the intersection of to the and the set of real35 valued 9directions. .using 34 The regularity assumption the intersection oftocontinuous and function , called heat associated 21 Dirichlet Forms esult has been progressed many Thiskernel article willfor focus theLaplacian 8 in function isoncalled the heat kernel associated the Laplacian such that is smooth all and there exists 36 metric. This is necessary or which for all 35 valued continuous functions with compact support on , is dense in under the uniform 35 valued continuous functions with compact support on , is dense in under the uniform 16 such that is smooth for all an 17 for which for all General heat equations are given by replacing the Laplacian with other heat operators. to the Laplacian. Less than sixty years ago, Nash (Nash, of the underlying spaces. See (Oshima et al, 2010) for profile satisfies some regularity assumptions. The 35 valued continuous functions with compact support on , is dense in und satisfies some regularity assumptions. The profile provided that the initial heat profile 7 satisfies some regularity assumptions. The 22 To simplify the analysis, the author assumes that the topolog perturbation of the heat equation using forms approach. 10 DirichletGeneral heat equations are given by replacing the Laplacian with other hea irichlet Forms . 37 functions, otherwise it will 36 metric. This is necessary. The domain of a Dirichlet form must contain enough continuous 36 metric. This is necessary. The domain of a Dirichlet form must contain enough continuous Dirichlet Forms question any researcher might ask that or not the heat kernel associated that 9 progressed .whether hen has been inis17 many directions. article will focus on the 18the result . 1958), Since then the result has been progressed inThis many directions. This article will on the 36 metric. This isnon-compact, necessary. domain ofofafocus Dirichlet formheat mustkernel enou Moser (Moser, 1961, 1964, 1967), and Aronson further treatment of regularity Dirichlet forms. for which for allThe mptions. The 23 compact but second countable Hausdorff, the 11 One question any researcher might ask is thattowhether or and not the associ for which forcontain allBorel is called the heat kernel associated to the Laplacian is called the heat kernel associated to the Laplacian thor assumes that the topological space is locally 38 et al, 2010) for further treat 37 functions, otherwise it will not reflect analytic properties of the underlying spaces. See (Oshima 37 functions, otherwise it will not reflect analytic properties of the underlying spaces. See (Oshima 8 function is called the heat kernel associated to the Laplacian To simplify the analysis, the author assumes that the topological space is locally operator exhibits the same behavior as that of the heat kernel associated to the Laplacian. ed to the perturbation of the heat equation using Dirichlet forms approach. 19 results related to the perturbation of the heat equation using Dirichlet forms approach. 37 functions, otherwise it will not reflect analytic properties of the underlying spaces eral heat equations are given by1971) replacing the Laplacian with other heat operators. 10 General heat equations are given by replacing the Laplacian with other heat Dirichlet Forms 12 heat operator exhibits the same behavior as that of kernel toregu the 24 has full support, the Dirichlet form on is .38 Since the result has been progressed inworked many directions. This article will focus on39isheat the (Aronson, 1967, 1968, independently on forms. Aand Dirichlet form (E,D(E)) strongly local if operators. 18 . Since then the result has been progressed in the many directions. This article w et al, 2010) for further treatment of regularity of Dirichlet 38 etthen al,non-compact, 2010) forBorel further treatment of regularity of Dirichlet forms. able and Hausdorff, the measure on always Aassociated Dirichlet form the Laplacian than sixty years ago, Nash (Nash, 1958), Moser (Moser, 1961, 1964, 1967), and Aronson compact but second countable and Hausdorff, the Borel measure on always 38 et al, 2010) for further treatment of regularity of Dirichlet forms. 20 on any researcher might ask is that whether or not the heat kernel associated to that 11 One question any researcher might ask is that whether or not the heat kernel associated to that 9 . 13 Less than sixty years ago, Nash (Nash, 1958), Moser (Moser, 1961, 1964, 1967), an the results analysis, the author assumes that the topological space is locally 25 admits the carre du champ operator. related to the perturbation of the heat equation using Dirichlet forms approach. 19deduced results related to the of the heat equation using forms approa for any with this problem which canthat bestrongly that the result 39 A Dirichlet form is strongly local ifif perturbation for any with 39 support, Aand Dirichlet form is strongly local for any with 40 for some con on is regular, local, and nson, 1967, 1968, 1971) independently worked on this problem which canLaplacian. be deduced that has full the Dirichlet form on is regular, strongly local, and 39 Abehavior Dirichlet form is strongly local if Dirichlet for any Dirichlet Forms 21 Dirichlet Forms or exhibits the same behavior as of the heat kernel associated to the 26 12 heat operator exhibits the same as that of the heat kernel associated to the Laplacian. 14 (Aronson, 1967, 1968, 1971) independently worked on this problem which can be d mpact, second countable and Hausdorff, the Borel measure on always snsare by the Laplacian with operators. aregiven given byreplacing replacing the Laplacian withother other heatfact operators. 20 40 for some constant .. heat The strong locality is equivalent to Leibnitz rule 4010 forauthor some constant The fact strong locality is equivalent to Leibnitz rule General heat equations are given byisisfor replacing theand Laplacian with other heat operators. holds for uniformly elliptic operators on i.e. for some constant . The fact is strong locality is 40 for some constant . The fact is strong locality is equivalent result holds for uniformly elliptic operators on i.e. for heat operators of the form admits the carre du champ operator. simplify the analysis, the assumes that the topological space is locally 22 To simplify the analysis, the author assumes that the topological space is locally ixty years ago, Nash (Nash, 1958), Moser (Moser, 1961, 1964, 1967), Aronson 27 Definition 1. A Dirichlet form is a pair where 13 oror Less than sixty years ago, Nash 1958), Moser (Moser, 1961, 1964, andoperators Aronson o ask isisthat whether not kernel to that might ask that whether notthe the heat kernel associated to(Nash, that 15heat result holds for uniformly elliptic operators onForms i.e. 1967), for heat dmight theoperators. Dirichlet form on isassociated regular, local, Dirichlet Forms 21the Dirichlet 11 One question any researcher might ask is strongly that whether orand not the heat kernel associated to that eat t non-compact, second countable and Hausdorff, the Borel measure on always 23 compact but non-compact, second countable and Hausdorff, the Borel measure on always 1967, 1968, 1971) independently worked on this problem which can be deduced that equivalent to Leibnitz rule and chain rule (Sturm, 1994). such that is smooth for all and there exists 28 positive symmetric, densely defined, bilinear form on with thet 14 (Aronson, 1967, 1968, 1971) independently worked on this problem which can be deduced that me behavior as that of the heat kernel associated to the Laplacian. me behavior as that of the heat kernel associated to the Laplacian. hamp operator. Toheat simplify the ofexhibits analysis, the author assumes thatthat thethe topological is locally irciated where is abehavior closed, 22 To simplify the analysis, thespace author assumes that spac operators the form such 16 such isis smooth for all the topological and heat the as that of heat kernel associated tolocal, the Laplacian. Definition 1. Aoperator Dirichlet form is a same pair where a closed, to12 that pport, and the Dirichlet form on is regular, strongly local, and 24 has full support, and the Dirichlet form on is regular, strongly and holds for uniformly elliptic operators on i.e. for heat operators of the form 29 any , the function and sh (Nash, 1958), Moser (Moser, 1961, 1964, 1967), and Aronson ash (Nash, 1958), Moser (Moser, 1961, 1964, 1967), and Aronson 15 with thetheyears result holds for (Nash, uniformly elliptic operators on i.e. for heatand operators of the form Hence, strong locality informally implies that thethe Dirichlet compact but non-compact, second countable andbut Hausdorff, thewith Borel onand always 23 compact non-compact, second countable Hausdorff, Borel measure near form following property: for 13on Less than sixty ago, Nash 1958), Moser (Moser, 1961, 1964, 1967), for which for all he Laplacian. positive symmetric, densely defined, bilinear form on the measure following property: for Aronson arre du champ operator. 25 admits the carre du champ operator. form is a pair where is a closed, independently worked on this problem which can be deduced that 17 for which for all )richlet independently worked on this problem which can be deduced that that is smooth for all i,j = 1, ...,n such that is smooth for all and there exists has full support, and the Dirichlet formhas on is regular, local, form’s behavior isform that offor differential operators. such that isproblem smooth all exists 30full The set equipped with the and inner product support, and the Dirichlet on is there regular, stron and 14 the (Aronson, 1967, 1968,.24 1971) independently worked on this which can beand deduced that and Aronson 26 any ,bilinear the16 function and .strongly Since then result has been progressed in many directions. This article will on the ly elliptic operators on i.e. for heat offollowing form mly elliptic operators on i.e. for heatoperators operators ofthe the form 3 1994). 1 focus and chain rule(Sturm, Hence, stron densely defined, form on with the property: for 18 . Since then the result has been progressed in many directions. This article will f admits the carre du champ operator. 25 equation admits the carre champ operator. 15 theperturbation result for uniformly elliptic operators on heat operators of the(E,D(E)) form Hilbert space, called ai.e. Dirichlet space, and norm associated to its inn forms Note that anybehavior regular Dirichlet form that 1.deduced A Dirichlet form pair where is a 2for closed, 27 Definition 1. Aaholds Dirichlet form iswhich a 31 pair where isthat aitsclosed, lts related to the the heat using approach. for which fordu all ed with the inner product isthere aDirichlet and there exists λof∈ (0,1) 17is forfor which for all isusing of differential operators. The set equipped with the product is aDirichlet nction and .inner 19 results related to the perturbation of the heat equation forms approach. 26 is smooth for allall and exists is smooth for and there exists 32 as the Dirichlet norm. of28 the form mmetric, densely defined, bilinear form onpair with the following property: for positive symmetric, densely defined, bilinear onis with following property: for2 exists Γwill which issmooth uniquely associated energy measure 16 such for all and there hen the result has been inthat many directions. This article will focus on theto isanisreferred nd its norm associated toaprogressed its inner product isDefinition referred toform Definition 1. A Dirichlet form is a where a closed, 3the Note that any regular Dirichlet 18 . Since then the result has been progressed in many directions. This article focus on the fo Hilbert space, called Dirichlet space, and its norm associated to its inner product to 20 27 1. A Dirichlet form is a pair where 33 equipped with the inner product is a for all 1 and chain rule(Sturm, 1994). Hence, strong locality informally implies that the Dirichlet form’s for all for all Dirichlet Forms ,as the function and . 29 any , the function and . to the perturbation of the heat equation using Dirichlet forms approach. dedthere exists ʹ might not be absolutely continuous with respected to 19 densely results related to positive the34 perturbation of the heat using Dirichlet forms approach. positive symmetric, defined, form on The withequation theassumption following property: forintersection the17 Dirichlet norm. 21 Forms measure which might not of betheabsolutely regularity means the and 28 bilinear symmetric, bilinear form on with followin for which forlocally all abeen space, and itsdirections. norm associated to its inner product isdensely referreddefined, to 4 Dirichlet 2Dirichlet behavior isthe that of differential operators. To simplify analysis, the author assumes that the topological space is been progressed in many This article will focus on the progressed in many directions. This article will focus on the 20 set equipped with the inner product is a any , the function and . 22 Toprogressed simplify the analysis, the that the topological space 30 The set then the equipped with, the inner product is aon 5 author Ifν.itIfassumes that always the volume measure ithappens happens that isis dense 35 valued continuous functions with compact support on in 29 any function and . ab Since the result has been progressed 18 .of Since result has been directions. article will focus the, is .tion the intersection and ,forms the set of real3 ofof Note that any regular Dirichlet form isDirichlet uniquely associated an energy on the heat using Dirichlet forms approach. the heatequation equation using Dirichlet approach. pact but non-compact, second countable and Hausdorff, the inBorel measure onThis always The regularity assumption means the intersection ofmany and , tothe setto of realDirichlet Forms 21 Forms 23 compact but non-compact, second countable and Hausdorff, the Borel measure o 36 metric. This is necessary. The domain of a Dirichlet form must con ce, called a Dirichlet space, and its norm associated to its inner product is referred 6 Dirichlet form is said to admit the carre 31 Hilbert space, called a Dirichlet space, and its norm associated to its inner product is referred to always absolutely continuous with respected to ν, then 19continuous results related toThis the perturbation of the heat equation using Dirichlet forms approach. The equipped with product isinner a . product lfull focus on the ʹset inand many directions. article will focus on thethe results ct support on , is dense inauthor under thethat uniform 30 The setinner equipped with the support, the Dirichlet form on is regular, strongly local, and 4valued measure which might not be absolutely continuous with respected to the volume measure functions with compact support on , dense in under the uniform simplify the analysis, the assumes the topological space is locally 22 To simplify the analysis, the author assumes that the topological space is locally 24 ofhas full37and support, and the form on toproperties regular, strongly functions, otherwise it will not reflect analytic ofexistence the underlyin yhlet the intersection , the setDirichlet 32 as Dirichlet the means Dirichlet norm. 7 product conditions for the of th h. Forms of5assumption anorm. Dirichlet form must contain enough continuous Forms Hilbert acountable Dirichlet space, and its norm associated toof itsrealinner is referred that Dirichlet form isSufficient said to admit the carre duischamp its the carre du champ operator. Hilbert space, called a must Dirichlet space, and its norm associated to its inner produc related tonecessary. the perturbation ofisHausdorff, the heat using If20Dirichlet itspace, happens that is31 always absolutely continuous with respected tocontinuous ,Borel then that metric. This iscalled The domain of38 aequation Dirichlet form contain enough ut non-compact, second and the Borel measure on always 33 23 compact but non-compact, second countable and Hausdorff, the measure on always 25 admits the carre du champ operator. et al, 2010) for further treatment of regularity of Dirichlet forms. nctions with compact support on , dense in under the uniform 8 Hirsch’s book (Bouleau and Hirsch, 1991) 21 Dirichlet Forms ysis, author assumes that topological space isis lysis, author assumes thatthe the topological space islocally locally alytic properties of the underlying spaces. See (Oshima asthe the Dirichlet norm. 32 the as the Dirichlet norm. functions, otherwise will not reflect analytic properties of Dirichlet the underlying spaces. (Oshima operator Sufficient 634 Dirichlet form isitmeans said to admit carre duDirichlet champ operator .. strongly regularity assumption the intersection of and , theform setlocal, of realpport, and the Dirichlet form on regular, strongly and See 26 The regularity assumption means the intersection of and , ,is the of realDirichlet forms approach. 24 has full support, and the form on issetregular, local, 39 A strongly local if fo essary. The domain of a Dirichlet form must contain enough continuous 9 Assuming the existence of theand carr 22 To simplify the analysis, the author assumes that the topological space is locally cond countable and Hausdorff, the Borel measure on always nition 1. A Dirichlet form is a pair where is a closed, econd countable and Hausdorff, the Borel measure on always 33 rity of Dirichlet forms. et al, 2010) for further treatment of regularity of Dirichlet forms. carre du champ operator. inuous functions with compact support on , is dense in under the uniform 27 Definition 1. A Dirichlet form is a pair where i 35 valued continuous functions with compact support on , is dense in under the uniform 7 Sufficient conditions for the existence of the care du champ operator is discussed in Bouleau and conditions for the existence of the care du champ 25 admits the carre du champ operator. 40 for some constant . The fact is strong locality is eq it will not reflect analytic properties of the underlying spaces. See (Oshima The regularity assumption means the intersection of and , the set of real10 champ operators, however, provide a way ichlet form on isisregular, strongly local, and 34 The regularity assumption means intersection ,t 23A Dirichlet compact but non-compact, second countable and the Borelthe measure on ofalways and richlet form on regular, strongly local, and Hausdorff, etive is locally rongly local if for any with symmetric, densely defined, bilinear form on with the following property: for form is strongly local if for any with 26 is is necessary. The domain of a Dirichlet form must contain enough continuous metric. This(Bouleau is of necessary. The domain of awill Dirichlet form must contain enough continuous 836 Hirsch’s book and 1991) and repeated here. 28Hirsch, positive symmetric, densely defined, bilinear withbook the following p rrator. treatment of has regularity Dirichlet forms. operator is.is discussed inform Bouleau and Hirsch’s valued functions with compact support onnot , be is dense the on uniform perator. 35 valued continuous functions with compact support on , is dense in a closed, und 24 full support, and the form on isinSee isunder regular, strongly local, and is on always The fact is strong locality is equivalent toDirichlet Leibnitz rule , continuous the function and 1. A Dirichlet form is aDefinition pair where a(Oshima closed, for some constant . The fact is strong locality equivalent to Leibnitz rule 27 1. A Dirichlet form is a pair where therwise it will not reflect analytic properties of the underlying spaces. 37 functions, otherwise it will not reflect analytic properties of the underlying spaces. See (Oshima 9 Assuming the existence of the carre du champ operator might seem strong. The carre du 29 any , the function and . orm is strongly local if for any with metric. This is necessary. The domain of a Dirichlet form must contain enough continuous 36 on metric. This isthenecessary. The domain form must contain enou 25 admits theregularity carre du champ operator. gly local, and mmetric, densely defined, bilinear form following property: forof ais Dirichlet for further treatment of of Dirichlet forms. et al, 2010) further treatment of ofadefined, Dirichlet forms. 1038 operators, provide a regularity way to define theto distance 28 positive symmetric, densely bilinear form on with the following property: for The with the inner a(Oshima rm isisachamp pair where iswith closed, orm aset pair where is aproduct closed, 26 me constant . for The fact isequipped strong locality is equivalent Leibnitz rule functions, otherwise ithowever, will not reflect analytic properties of the underlying spaces. See 30 The set equipped with the inner product 37 functions, otherwise it will not reflect analytic properties of the underlying spaces called intrinsicwith distance on . This tie Santi Tasena /property: Burapha Sci. 19 1 11 : 169-177 170Definition ,space, the function . (2014) irichlet form strongly local ifand for any with 39 Aform form is strongly local if J.inner for any the 29 any , following the function and . ert called a Dirichlet Dirichlet space, and its norm associated to its product is referred to efined, form onon is with the property: for defined, bilinear with the following for 27 1. A Dirichlet form is a pair where is a closed, et al,bilinear 2010) for further treatment of regularity of Dirichlet forms. 38Hilbert et al,space, 2010)called for further treatment of regularity of Dirichlet forms. 31 a Dirichlet space, and its norm associated to its inner product is 12equivalent space to. Leibnitz In orderrule to define the intrinsic for some constant . The fact is isifequivalent towith Leibnitz rule 40 for some constant The fact isform strong and . . .locality eset Dirichlet and equipped with the inner product isstrongly a product Dirichlet form isstrong strongly local foris any with is a closed, 30 symmetric, The equipped the inner is a 28 Anorm. positive densely defined, onlocality with the following property: for 39set Abilinear Dirichlet form is local if for any 32 as the Dirichlet norm. 13 continuous with respect to the volume mea ce, a Dirichlet space, and its norm associated its innerlocality product isequivalent referred totoofLeibnitz 33on called the intrinsic distance . This tiesfact thetoisDirichlet form toisthe geometry the underlying 11 called for some constant . The strong rule 3 4 5 2020 see see (Sturm, 1995a) 1995a) and and(Hirsch, (Hirsch, 2003) 2003) for the proof proof of of this this fact. fact. on It is is well-known (Oshima et al, 2010) there and only one form 24 on . the Itfor isthe well-known (Oshima etthat al, and 2010) there is one andDirichlet onlyet one Diric 14(Sturm, the domain . inTherefore, existence ofoperator the du champ operator is natural. 20 (Sturm, 1995a) (Hirsch, 2003) for the proof of this fac 24 on Itisthat isone well-known (Oshima al, 2010 e and a heat operator i.e. 24 a nonpositive, densely defined, self-adjoint 19 between any twoHence, points actually the of aiscarre shortest path between those two points, 3 see 23 Let be length a .Equation heat operator i.e. apath nonpositive, dense chain rule(Sturm, 1994). strong locality informally that the Dirichlet form’s 19 between any two points inimplies actually the of ainshortest between those 22 Heat Equation 22 22 Heat Heat Equation Equation 22length Heat Equation 22 Heat 22 Heat Equation 21 21 25 on such that is dense in under the Dirichlet norm, and such that is dense under the Dirichlet n 25 on 21in 15et(Sturm, A common assumption of24 heat analysis it must be a complete 25kernel onis that such that -known (Oshima al, 2010) that there issee one and only one Dirichlet form 20is that see 1995a) and (Hirsch, 2003) for the proof of this fact. onthe .operator It isbe well-known (Oshima etnonpositive, al, 2010) thatdensely there isdis behavior of differential operators. 20 (Sturm, 1995a) and (Hirsch, 2003) for the proof of this fact. 23 Let be a heat operator i.e. a nonpositive, den 23 232222 Let Let be be a a heat heat operator operator i.e. i.e. a a nonpositive, nonpositive, densely densely defined, defined, self-adjoint self-adjoint operator operator 23 Let be a heat operator i.e. a nonpositive, 23 Let be a heat i.e. a nonpositive, densely defined, self-adjo 23 Let a heat operator i.e. a Heat Equation 26 forHeat all22Equation . Assuming Therefore, studying heat operators and Dirichlet for . Therefore, studying heat Equation operators Heat 16 metric and26itinmust metrise the topology ofall 26 . and metrises the for topology of isin. and all The such that is dense under the Dirichlet norm, 21 25 on such that isand dense Note any Dirichlet form is that uniquely associated to2010) an energy 21 24 on . It is well-known (Oshima et al, 2010) that there 24 242323 on on that informally . . Itregular Itisis well-known well-known (Oshima (Oshima et et al, al, 2010) 2010) that there there is is one one and and only only one one Dirichlet Dirichlet form form 24 on . It is well-known (Oshima et al, 2010) that there 24 on . It is well-known (Oshima et al, that there is one only one D 24 on . It is well-known (Oshima et al, 2010) that there is one Let Let be be a a heat heat operator operator i.e. i.e. a a nonpositive, nonpositive, densely densely defined, defined, self-adjoint self-adjoint operator operator 27 forms are complementary. 27 forms are complementary. nce, strong locality implies that the Dirichlet form’s 23metric heat operator areplace nonpositive, de 17 necessary but3 studying assume itheat is a complete is redundant since itbe isaall always possiblei.e. 27Let forms are for complementary. for all 22ʹ . Therefore, operators and Dirichlet Heat Equation 26 . toTherefore, studyin 22 Heat Equation measure which might not be absolutely continuous with respected to the volume measure . 25 on such that is dense in 25 25 on on such such that that is is dense dense in in under under the the Dirichlet Dirichlet norm, norm, and and such that is dense in 25 on such that is dense in under the Dirichle 25 on 25 on such that is dense in un Itisiscompletion. well-known well-known (Oshima (Oshima et etal, al, 2010) that that there there is only only one one Dirichlet Dirichlet form form 28 Abe heat equation heat operator isdensely nothing a generalization of the2010) classical rators. 2424 28 Aoperator heatwith equation with heat operator isgeodesic nothing but a with generalization of theheat 24 on . isone Itone is well-known (Oshima et al, that there 18 with Under these assumptions, isand aand space i.e. the distance 28 Abut heat equation heat operator isclas no ary. 23 onon Let. . Itits aallheat a2010) nonpositive, defined, self-adjoint operator 27 forms are complementary. Let be adense heat operator i.e. aall nonpositive, densely defined, self-adjo If it26 happens that and is2923 always absolutely continuous with respected to ,for then that 26 all . Therefore, stud 2625 for for all .i.e. . energy Therefore, Therefore, studying studying heat heat operators operators and and Dirichlet Dirichlet 26 for all . Therefore, stu 26 for all . Therefore, studying heat operators (Bouleau Hirsch, 1991) and will not be repeated here. equation associated to L is a function u such that 26 for . Therefore, studying ha 1 and chain rule(Sturm, 1994). Hence, strong locality informally implies that 25 on on such such that that is is dense in in under under the the Dirichlet Dirichlet norm, norm, and and 29 equation with the Laplacian . Informally, a solution of the heat equation associated to is a equation with the Laplacian . Informally, a solution of the heat equation associated richlet form is uniquely associated to an 25 2010) such that is dense 19 between any twoa points in is actually the length ofon aisshortest path between those.is two points, equation with the Laplacian Informally, aainsol n with heat operator is nothing but generalization of the classical heat 24that on . form’s Itadmit is well-known (Oshima et al, that there one and only one Dirichlet form 28 A29 heat equation with heat operator nothing but ge 24 on . It is well-known (Oshima et al, 2010) that there is one and only one Di 27 forms are complementary. 27 272626 forms forms are are complementary. complementary. y implies the Dirichlet 27 forms are complementary. Dirichlet form is said to the carre du champ operator , . 27 forms are complementary. 27 forms are complementary. 2volume behavior that differential operators. for for allall .is .that Therefore, Therefore, studying heat operators and and Dirichlet 20 Assuming the existence of the carre du champ This can beoperators interpreted as Dirichlet 30 function such that .ofdense beheat interpreted as 301995a) such This can beDirichlet interpreted as.and bsolutely with respected tofunction the measure .isthe see (Sturm, and (Hirsch, 2003) for proof of..the this fact. 26 for Therefore, stud 30 function such that .theThis can acian .25continuous Informally, a solution of the heat equation associated toThis isstudying acan on such that in under the norm, 29 equation with Laplacian .all Informally, aunder solution ofof the such that is dense in Dirichle 25 on 28 A heat equation with heat operator is nothing but ahca 28 282727 conditions A A heat heat equation equation with with heat heat operator operator is is nothing nothing but but a a generalization generalization of of the the classical classical heat heat 28 A heat equation with heat operator is nothing but 28 A heat equation with heat operator is nothing but a generalization the Sufficient for the existence of the care du champ operator is discussed in Bouleau and 28 A heat equation with heat operator is nothing but a genera 3 Note that any regular Dirichlet form is uniquely assoc forms forms are are complementary. complementary. 31 ǡ ሺ ǡ ሻ for all test functions . However, what are test functions? 31 ǡ ሺ ǡ ሻ for all test functions . However, what are test functions? 21 27 forms are complementary. continuous with respected to , then that operator might seem strong. The carre du champ for all test 31 ǡ ሺ ǡ ሻ for all test functions . H atlways absolutely . This can be interpreted as 26associated forequation all .Laplacian Therefore, heat Dirichlet 30 function such that .and This be interpret 26 for all . Therefore, studying operators an 29 equation with the Laplacian .the aequation of the uniquely to an energy 29 292828 equation equation with with the the Laplacian Laplacian . .heat Informally, Informally, aequation abe solution solution of of the the heat heat equation equation associated associated toto isisheat acan aheat 29 equation with the Laplacian . Informally, Informally, asolution solution ofHil th 29with with the . studying Informally, a.,operators solution of the heat associa Hirsch’s book (Bouleau and Hirsch, 1991) and will not repeated here. ʹ 29 with the Laplacian Informally, a solution of the heat ea A A heat heat equation equation with heat operator operator is is nothing nothing but but a a generalization generalization of of the classical classical heat 4 measure which might not be absolutely continuous with respected to the 22 Heat Equation 32 Given a Hilbert space and an open interval denote the Hilbert space 32 Given a Hilbert space and an open interval , denote the 28 A heat equation with heat operator is nothing but the carre du champ operator , . for all 27 testoperators, functions .complementary. However, whatforms areway test functions? 32 Given a Hilbert space and anbe open inter forms arehowever, ν . However, what are test functions? provide a to define the functions 31 ǡ ሺ ǡ ሻ for all test functions . However, what 27 are complementary. 30 function such that . This can interpr 30 30 function function such such that that . . This This can can be be interpreted interpreted as as function that . can can be respec interp Assuming the existence of carre du champ operator might seem strong. The carre du function such that .such can be interpreted espected to the volume measure .30the 30 function such that .functions be interpreted 29 equation equation with with the the Laplacian . Informally, a30 aand solution solution of ofbut the the heat heat equation equation associated associated to toThis isis aaas aoperator 5.measurable Ifoperator it operator happens that is always absolutely continuous with 23 Let aInformally, heat i.e. awith nonpositive, densely defined, self-adjoint 33 of allLaplacian measurable functions finite norm 33 of allbe functions with finite norm 29 equation with theThis Laplacian . This Informally, solution of thea ence of29 the care du champ operator is, discussed in Bouleau 33 of all measurable with finite space an open interval denote the space 28and A heat equation with heat isHilbert nothing aaare generalization of the classical heat 32 Given Hilbert space and an open interval , den Given a Hilbert space and an open interval distance 28 A heat equation with heat operator is nothing but a generalization of the cl 31 ǡ ሺ ǡ ሻ for all test functions . However, wha 31 313030 ǡfunction ǡfunction ሺ ሺ ǡ ǡ ሻ ሻ for for all all test test functions functions . . However, However, what what are test test functions? functions? champ operators, however, provide a way to define the distance 31 ǡ ሺ ǡ ሻ for all test functions . However, us with respected to , then that 31 ǡ ሺ ǡ ሻ for all test functions . However, what are test functions? ǡ acan ሺthat ǡ ሻheat for all testand functions .This However, what arewh te such such that .31 . (Oshima This This30 can be interpreted interpreted asequation as 6 . Informally, Dirichlet form isbe2010) said the carre duonly champ operator 24 on It is well-known etsolution al, there isthat one one Dirichlet form function such . to be interp h, 1991) 29 and will not finite be with repeated here. ions with norm equation the .that Laplacian ofto theadmit associated is can anorm 33 of all measurable functions with finite 29 equation with the Laplacian . Informally, a solution of the heat equation associat denote the Hilbert space of all 32 Given a Hilbert space and an open interval , d,H 32 32 Given Given a a Hilbert Hilbert space space and and an an open open interval interval , , denote denote the the Hilbert Hilbert space space tor , . 32 Given a Hilbert space and an open interval 32 Given a Hilbert space and an open interval , denote the 32 Given a Hilbert space and an open interval , denote 31 31 ǡ ǡ ሺ ሺ ǡ ǡ ሻ ሻ for for all all test test functions functions . However, . However, what what are are test test functions? functions? 25 on that that dense inሺ ǡasሻ ofunder the Dirichlet norm, andis discu all 31 ǡ is the existence for testdu functions . However, wh 7 such Sufficient conditions for the care champ operator f the carre operatorsuch might seem strong. The carre du 30du champ function . such This can be interpreted 30 function that . This can be interpreted as 33 of all measurable functions with finite norm 33 33 of of all all measurable measurable functions functions with with finite finite norm norm 33 of all measurable functions with finite norm 33 space of 8alland measurable functions with finite norm perator is32 discussed in Bouleau 33 ofbook all functions with finite norm measurable with finite norm Given Given aHilbert Hilbert and an anopen open interval interval ,and denote , functions denote the the Hilbert Hilbert space space 26ǡthe for all . Therefore, studying heat operators and Dirichlet Hirsch’s (Bouleau Hirsch, and will not repeated here. , e a way32 to distance 32 Given aare Hilbert space and an be open interval 31define aLet ሺand ǡ31ሻspace for test functions .measurable However, what test1991) functions? ǡ all ሺ ǡ ሻ for all test functions . However, what are test functions? 34 be the set of all functions whose distributional derivative can s Let 34 be the set of all functions whose distributional derivat eated here. Let 34 be the set of all functions called3333 the intrinsic distance on . This ties the Dirichlet form to geometry of the underlying of of all all measurable measurable functions functions with with finite finite norm norm 27 forms are complementary. 9 Assuming the existence of the carre du champ operator might seem 33 of all measurable functions with finite norm 32 Given a Hilbert space and an open interval , denote the Hilbert space Given space and an open interval denote H 35 beArepresented asdistributional adistance, function in Equip with the norm 35 be represented asaitoperators, aHilbert function in . Equip with the normheat. Equip be rset might seem The carre du32intrinsic classical 35 represented as afunctions function in the of all whose derivative can space . functions Instrong. order to define the is necessary that is absolutely 28 heat equation with heat operator is .nothing but a generalization of,the thedistance 10 champ however, provide aset way to define Let 34 be the of all whos 3 33 of all measurable functions with finite norm 33the with of allthe measurable functionsainsolution finite norm of associated to is a e continuous ction inties thewith . Equip norm to the volume measure forInformally, functions a sufficiently subset 29 respect with Laplacian of the large heat equation distributional awith 35 beLet represented as function in . Equip w This34 Dirichlet form tothe the geometry ofties the 34 be set of wh Let Let 34 be be the set set of ofXall all functions functions whose distributional derivative derivative can can . This the.underlying Dirichlet Let bethe the set of all functions called the equation intrinsic distance on Let 34the be the set ofall allfunctions functions w Let 34 be set ofwhose allbe functions Let 34 the set of all functions whosederi dis whose distributional the domain . Therefore, the existence of the carre du champ operator is natural. 30 function such that . This can be interpreted as intrinsic isgeometry necessary that is the be as anorm function inin 35 353434distance, be be represented represented as asaabe function function inin .absolutely .Equip Equip with with the berepresented represented as anorm function .can Equip 35 be represented as arepresented in as . the Equip thebe norm 35 be a itfunction in aimplies . Equip th form to1 itthe ofof the space whose distributional derivative can X35 .35 Infunction . Equip Let Let the the set of ofሻthe all all functions functions whose whose distributional distributional derivative derivative can Let 34 be of all functions and rule(Sturm, 1994). Hence, strong locality informally that the with Dirichlet form’swithwh A common assumption heat kernel analysis is that must be complete 31 ǡ chain be ሺsetin ǡunderlying for all test functions . However, what areset test functions? 11 called the intrinsic distance on . This ties the Dirichlet form to the geomet ume measure for functions in a sufficiently large subset of makes 36metrise Hilbert makes 36 a Hilbert space. 35 be represented represented asGiven as athe ais function function in all . Equip .space. Equip with with norm norm ofinin derivative makes 36 whose a is Hilbert space. represented asthe atopology Equip order to the intrinsic itof isafunctions necessary that 35 be represented asthe afunction function .. Equip 2 define behavior that of in differential metric and itbe must topology . operators. Assuming metrises the Let 34 be the set of distributional can spacethat to the35 geometry of the underlying 32 adistance, Hilbert space and open ,the denote the 12 space .bean In order intrinsic distance, it isHilbert necessary Let 34 the set interval oftoalldefine functions whose distributional deriv existence of the carre du champ operator is natural. Hilbert space. 3 Note that any regular Dirichlet form is uniquely associated to an energy necessary but assume it is a complete metric is redundant since it is always possible to replace with the norm is absolutely continuous with respect to the 36 a Hilbert space. makes essary that is absolutely 35 be represented as a function in . Equip with the norm 33 of all measurable functions with finite norm 2 complete continuous with respect to the volume measure for functions in a sufficie suffici represented as a function in . Equip with the norm f in the heat kernel analysisʹ is35that be it 213 must be a 2 This means This means for all for all. . This means arespected Hilbert completion. Under these assumptions, amakes geodesic i.e. the distance 4 measure measure which not domain be of absolutely with tothe the volume measure . for all s with in36 aits sufficiently large subset of ν for functions fmight in atopology sufficiently large volume 36 space. 36 aametrises Hilbert Hilbert space. space. makes makes 36isaisHilbert a Hilbert space. makes 14 the . continuous Therefore, the existence of carre du champ operator is 36 space.space makes topology of . Assuming the 36 makes a Hilbert space. 2 ans for all . between any two isis actually the length a shortest path between those two points, 5 points Ifsince itinhappens that 15 isof always absolutely continuous with respected tofor , then isthat This of means all . it m mp operator isredundant natural. A common assumption in the heat kernel analysis that lete metric is it always possible to replace subset of the domain D(E) . Therefore, the existence of 3636 makes makes and (Hirsch, a 2003) aHilbert Hilbert space. space. 36 fact. a Hilbert space. makes 2 operator 2 seeis(Sturm, for the to proof offor 2 2this 6 be Dirichlet form isspace said admit the carre dumetrise , forforallallmetrises . .th. ysis that it1995a) must ais2complete 2champ This means This This means means for all all . . 16 metric and it must the topology of . Assuming This means se assumptions, a geodesic i.e. the distance This means for all . This means for all . carre du champ operator is natural. Let 34 be the set of all functions whose distributional derivative can 36 the a Hilbert space. makes metrises the topology of is 7 Sufficient conditions for the existence of the care du champ operator is discussed in Bouleau and 36 makes aallHilbert space. 2 2 17 those necessary but it is metric is redundant4since it is always .p 2 ctually the length of a shortest path between two points, This This means means for for allassume . . a complete This with means for all ρ in and the kernel a Hilbert space. possible assumption 35 8A common beto represented as(Bouleau aoffunction inheat . makes Equip the norm Heat Equation e it is always replace Hirsch’s book Hirsch, 1991) and will not be repeated here. 18 with its completion. Under these assumptions, is a geodesic spa 003) for the proof of this fact. 2 Letspace beitadistance heat i.e. a nonpositive, densely defined, self-adjoint operator This means for all Note .thatoperator Note analysis9i.e. is that must beoperator a complete and121itofThis must so Note that that so so geodesic the Assuming the existence the carre du champ seem strong. The carre means formight all the .of a shortest 19 metric between any two points in is actually length pathdubetwee 2 on . It is well-known (Oshima et al, 2010) that there is one and only one Dirichlet form soof 10 operators, however, provide aTherefore, way tothat define the distance est path between those two 22 the Therefore, isis a,a metrise thechamp topology of X. Assuming metrises the -dual D(E) . this Therefore, 1 Heat Note that points, so ,,2003) the Lfor -dual of . fact. 20 1ρsee (Sturm, 1995a) and (Hirsch, the proof of 1 Note that Note Equation on such that is dense in under the Dirichlet norm, and t.erator2 i.e.Therefore, 3operator 3 that is a1 well-defined a topology nonpositive, self-adjoint X isdensely necessary but1assume aNote complete 2 object. Therefore, 2it is space. Therefore, wel1defined, Note that so so Note 4a-dual that Denote , the,is4 the so 36 ofmakes a21 Hilbert for all . Therefore, studying heat operators and Dirichlet 4 4 the the set set of of all all functions functions such such that that for fo 22Therefore, Equation shima3et al, 2010) there issince one2itand one form is a well-defined and denote object. Denote 3Heatalso 3 Dirichlet Therefore, 2only is a iswell-defined a well-defined obje metric isthat redundant is always possible to replace 2 Note Therefore, 1 Note that so , the -dual of forms are complementary. 1 that so 55.Let subset subset of of ,form ,there there exists anonpositive, function function satisfy satisf 23 be afor heat operator i.e. geometry aasupport densely 2functions is 11 densethe in set the under Dirichlet and called intrinsic distance This ties the Dirichlet toexists ofa.e. thesuch underlying 4 ofUnder all such that compact 4the set that of defined, allfor funct 4 onnorm, the set of allany functions any has a compact foropen 3 operator 3themeans and and alss This for all . relatively X with its completion. these assumptions, ( X,ρbut ) is 3to A 1heat equation with heat is nothing a generalization of the classical heat 2 Therefore, is a well-defined object. Denot nsely 5defined, self-adjoint operator 2 Therefore, 6 6 In In order order to view view a a solution solution of of heat heat equations equations as as a a function functio 24 It well-known (Oshima 2010) that there isrelatively one and only 1 12 Note that that so so.set ,a.e. ,al, the the -dual -dual of . relativel . exis . Therefore, studying operators and Dirichlet space In order toa function define the intrinsic distance, it is necessary that isof absolutely subset ofNote ,. there exists satisfying on . for subset of ,satisfying there 5 on subset of , there exists a5et function 1isset Note that 4heat 4 the set the allis of functions all functions that any for any com and also denote the ofof allthat functions 4 be the set all functions the Laplacian . distance Informally, a solution ofof the heat equation associated toaasuch asuch a2geodesic i.e.view the between any two points isequation one6 and 2with only one Dirichlet form 3space and also denot 7 7 will will be viewed viewed as as function function . . 3 25 on such that is dense in under the 4 13 continuous with respect to the volume measure for functions in a sufficiently large subset of Therefore, Therefore, is is a a well-defined well-defined object. object. Denote Denote In order to a solution of heat equations as a function from to , any function 6 In order to view a soluti 6 In order to view a solution of heat equations as a function from 5 subset 5 subset of of , there , there exists exists a function a function satisfying satisfying a.e. ona. 2 any Therefore, of , there exists a function Note that4 so , the 88 interpreted function such .a 1shortest This be as for5 all subset such that for any relatively compact under the norm, and 4that the set ofcan all the functions such that for relatively compact ope in3 nothing XDirichlet is actually of path between the set of all functions s 26 . Therefore, studying heat oper operator but athegeneralization of the classical heat 14 the domain . Therefore, existence of the carre du champ operator is natural. 7 will belength viewed as a function . 3is and and also also denote denote 7 will be viewed 7 will be viewed as a function . 6 functions 62 In, .there order In 9order to view toa function view a solution atest solution of heat ofbeequations heat equations as function as a function from from towell-defin to ,as an 3 a to 6 In order view a solution of heat e 9 Definition Definition 2. 2. Let Let be an an open open time time interval interval and and be be a a heat heat op o Therefore, is a ǡ ሺ ǡ ሻ for all test However, what are functions? 5 subset of exists satisfying a.e. on . ying heat operators and Dirichlet 8 4 complete 8isrelatively of thereit.exists exists aabe function open subset 8 forms 5 heat subset offorany ,,that there function 27 are complementary. nformally, 4athose of the see heat equation associated to isviewed 15 A common assumption of2003) inabe the kernel analysis must a two points, (Sturm, 1995a) and (Hirsch, 4solution the the set setofof allall functions functions such such that that for any relatively compact compact open open 7 7 will be will viewed as a function as a function . 4will the set of the al 7 an be viewed as aany function 10 10 local local regular Dirichlet Dirichlet form form on on .generalization function thata16 so8 3open ,equations the -dual of .the Definition Let be open time interval and be aLet heat operator associated to asolution strongly Given Hilbert space and interval denote the space 62. tometrise view aaDefinition of heat as aHilbert function to ,.A functio 9metrises Definition 2.ofbe Let be an 9 2. be open time interval and a..A heat operato 128 Note 6regular In order toso view afrom equations a A heat equation with heat operator is nothing but aheat 1 ,that that so . 9Note This can be interpreted as metric and itorder must the topology of .Note Assuming topology of is,open 8an satisfying a.e. on 5 for 5 the subset subset of of In ,an there , there exists exists function asolution function satisfying satisfying a.e. a.e. on on 8 1 1 Note Note that that so so proof of this fact. 5 subset of , the 11 11 solution solution of of the the heat heat equation equation if if generalization of the classical heat 4 the set of all functions such that for any relat of all measurable functions with finite norm 10 local regular Dirichlet form on . A function is a (local) weak 7 will be viewed as a function . erefore, is a well-defined object. Denote local regular Dirichlet form 10is local regular on .any A function will be viewed asabe function Therefore, isheat well-def equation with the Laplacian . view Informally, ato solution the equation 9test Definition 9 a229 Definition 2.complete Let 2. Let beequations an open anIn open time interval time interval and be aheat heat be heat operator associated associat 17 necessary but assume itsolution aso metric is redundant since it10 is and always possible to replace st functions what areto functions? 2Therefore, Therefore, 6 6. However, Inorder order toview view asolution of ofheat equations as asa9form afunction function from from to ,aoperator any ,of function function Definition 2. an open interv 78beDirichlet order to athe solution of as a atime 212 2heat Therefore, istoia 8theIn 6Let In to view 1 equation Note that , function -dual ofequations .order 12 a) a) , , and and heat11 associated to is a 5 subset of , there exists a satisfying solution of heat equation if and also denote 11 solution of the heat equation 11 solution of the heat equation if function such that 10 This can beassociated interpreted 10 bebe 10 local regular Dirichlet Dirichlet form form .A function . space A function is a (li 330local itswill completion. Under these assumptions, is on aa.regular i.e. the distance 7interval 7 18 9 with will viewed viewed as as aregular afunction function . on . local nd an2 open , denote the Hilbert space Dirichlet form 3b) Definition 2. Let be an time interval and be heat to aasand strongl 3open 7 will be on view function D(E) , geodesic any function issolution aI towell-defined object. Denote 9any Definition 2.ofcompact Let beoperator an openas time interval 13 133order b)for for any open open interval interval relatively relatively compact compact in and and any any 6 In to view afrom heat equations ainfunction from tob a) , and eted12with as Therefore, the set of all functions such that for any relatively open Heat Equation 12 a) , and 8 8 12 a) , and 11 11 solution solution of the of heat the equation heat equation if if 4 the set of all functions such that for any rela 31 ǡ ሺ ǡ ሻ for all test functions . However, what are test funct 19 between any two points in is actually the length of a shortest path between those two points, finite norm 11 solution of the heat equation if 4 the set of all functions such 8 10 local regular Dirichlet form on . A function is a (local) wea 4 4 the the set set of of all all functions functions such such that tha 10 local regular Dirichlet form on . A 3 and also denote will be viewed asrelatively afunction 7 532 will be viewed as aa.e. . a,strongly b)9of9for Definition any interval relatively compact in and any ,function at are13test functions? Definition 2.2.be Let Let be bean anopen open time interval interval and and be be a exists aheat heat operator operator associated associated to toaany strongly bset , open there exists aa heat function satisfying .exists 13 b) forinDefinition any open interval b) for interval compact and 12 12 a) a)13 , any and , subset and subset of , there a,on function satisfying Let (L,D(L)) operator i.e. atime nonpositive, 20 seeset (Sturm, 1995a) and (Hirsch, 2003) for the proof of this fact. Given aopen Hilbert space and an open interval denote 9acan 2. Let relativ besati an 12 a) , and 5subset of , there a function 11 solution of the heat equation if 5 5 subset of of , there there exists exists function a function s Let be the of all functions whose distributional derivative 8 11 the heat equation 4In order the set space of 13 all such that for relatively compact open 1010 tolocal local regular Dirichlet form on .solution A .In A function isof isaas a(local) (local) weak weak view aregular solution of functions heat equations function from to ,of any enote thedefined, Hilbert Ib) befor an open time interval and L if 2allas Definition 2.compact Letofany 633 Ina6).on order to view afunction solution heat equations aequations function from 13 b)form for b)any for open any open interval interval relatively relatively compact in and in any and any ,be 21 of measurable functions with finite norm 10 local regular Dirichlet 13 any open interval relatively compac 6is order to view afunction solution heat equations as afoft Dirichlet densely self-adjoint operator on L (X, ν It 9 Definition 2. Let be an open time interval and be a heat operator 12 a) , and 6 In In order order to to view view a solution a solution of of heat heat equations as, as a functi a asso fun be 5represented as a function in . Equip with the norm 12 a) , and subset of , there exists a function satisfying a.e. on . 14 14 1111 solution of of the the heat heat equation equation if if willsolution be viewed as a function . 7 local regular7Dirichlet will be Equation viewed as abe function . the heat equat 22 Heat a heat operator associated to as strongly local regular of will be viewed as a function 10 that form on .solution function b) open in and any ,Afunction well-known (Oshima etany al,a, 2010) there isrelatively one 7 7 and compact will will be viewed viewed as aa11 function a,relatively function . . a. 13asSimple b) for any open interval compact in and 6 In order to for view solution of heat equations a nonpositive, function from to any 15 15 Simple examples examples of of weak weak solutions solutions in in the the sense sense introduce introduc 8interval 1212 a)23 a)13 and , and unctions whose derivative can 8 2 14 Letdistributional be a heat operator i.e. a densely defined, self-adjoint operator 14 14 8of8athe , and 2 Dirichlet form (E,D(E)) L12(X, νa)). A function 11 solution heat equation ifaon efinition 2. Let b) beDirichlet an open interval and be heat operator associated to strongly (E,D(E)) on L15 (X, ν) compact such that only one form 7 . Equip will be time viewed aswell-known a solutions function an 9 Definition 2. Let be.any open time interval and be a and heat operator 16 16 for for ,are ,be where where istime is any any bounded bounded interval, interval, and 1313 b)for for any any open open interval interval relatively relatively compact in in and and any , , 9 Definition 2. Let an open time interval and be aash with the norm Simple examples of weak in the sense introduced above functions 15 15 Simple examples of wea Simple examples of weak solutions in the sense introduced abo 24 on . It is (Oshima et al, 2010) that there is one and only one Dirichlet form 9 9 Definition Definition 2. 2. Let Let be be an an open open time interval interval and be be a heat aand hea 13 b) for any open interval 14 14 8 1210 a)local , and 14 is a (local) weak solution of the cal regular Dirichlet form on . A function is a (local) weak regular Dirichlet form on . A function 17 17 examples examples are are given given in in (Aronson, (Aronson, 1968), 1968), see see also also (Gyrya (Gyrya & & SaloffSaloff Let 34under beregular the set of ,Dirichlet all whose D(L)25 ⊂ D(E) is dense D(E) theSimple Dirichlet 10local local regular form . functio func 16 for ,inwhere isSimple any bounded interval, and .isunder More interesting 16 for ,A where 16 for where any bounded interval, and such that is dense infunctions the norm, and 10 10open local regular Dirichlet Dirichlet form form on onon .above A .distribution A function functio 15on 15 examples examples of weak of weak solutions solutions inassociated the in sense the introduced introduced above are are makes a2.Hilbert 9 of the Definition Let space. be an open time interval and beNote aequation heat operator to aDirichlet strongly 15 Simple examples of weak solutions b) for any interval relatively compact in sense and any , fu lution heat equation if131135 ose distributional derivative can 14 heat if solution of the heat equation if 18 18 Note that that fixing fixing an an open open set set and and replacing replacing 14 be represented as a function in . Equip with the norm 17 examples are given in (Aronson, 1968), see also (Gyrya & Saloff-Coste, 2011). 11 solution of the heat equation if 17 examples given in (Aronson 17 examples are given in 1968), see also (Gyrya & Saloff-Cost norm, for all .solution 26and Dirichlet16 for all . ofTherefore, Dirichlet 16 1111 solution the the heat heat equation equation ifand ifare for for ,Awhere , of where is(Aronson, any isstudying bounded any bounded interval, interval, and . More . bM 10the norm local regular form .15 function is aoperators (local) weak forheat ,the where is any ,Note and 15 that fixing Simple examples weak solutions in 16 the sense introduced above are functions with 12 a) onof , and 19 19 in in the above above definition definitio Simple examples of weak solutions in the sen a) , and 12 a) , and 18 1414 an open set and replacing with the closure of the set 227 18 Note that fixing an op 18 Note that fixing an open set and replacing with forms are complementary. 12Dirichlet 12 given a) (Aronson, a) , see and , and 17compact 17 examples examples are are in (Aronson, 1968), also see(Gyrya also (Gyrya & Saloff-Coste, &(Aronson, Saloff-Coste, 2011). 2011). D(L) .This Therefore, heat operators solution of16the heatstudying equation if,given means forand alland . isinany 17 examples are14 given in 1968), see any open interval relatively in any ,1968), for where bounded interval, and . More interestin 13 b) for any open interval relatively compact in and any , 20 20 solutions solutions (in (in ) ) of of heat heat equations. equations. e.for11 16 for , where is any bounded in 19 in the above definition yields the definition of local 13 b) for any open interval relatively compact in and any 1515 28 Simple Simple examples weakNote solutions solutions in in the sense sense introduced introduced above above are are functions functions 19 19 in thethe above definition yie Aexamples heat equation with heat is nothing but generalization of classical heat clo b)the for any open interval relatively compact inwith and 13 13operator b) b) for for any any open open interval interval relatively relatively compact compact in inand and any any 18 18 ofofweak Note that fixing that fixing an open an open set setaand replacing and 15 replacing with the closure the Simple examples 12 a) , and 18 Note that fixing an open set forms are complementary. 21 21 17 examples are given in (Aronson, 1968), see also (Gyrya & Saloff-Coste, 2011). 14 17 examples are given in (Aronson, 1968), see also (Gyry 20 16 solutions )for of heatwith equations. 16 for , where , where is isany any bounded bounded interval, interval, and . More .associated More interesting interesting 20 solutions (in ) oftoheat 20 solutions (in equations. 29 (in equation the .22 Informally, solution ofand the heat equation isequation athe 19 1915Laplacian in, the in above the above definition definition yields yields the definiti de any 16 for 13 for any open interval relatively compact in and any) aaofHilbert 19 22 Definition Definition 3.heat 3. An An associated associated heat heat kernel kernel ofopen of aaintroduced strongly strongly local local regula regul Simple examples of weak solutions in the sense above ar 18 Note that fixing an open set and replacing with the closure of the se 21 b) L is nothing A heat equation with heat operator 36 makes space. 21 21 18 Note that fixing an set and repla for all . 1717 examples examples are aregiven given in in (Aronson, (Aronson, 1968), 1968), see see also also(Gyrya (Gyrya & & Saloff-Coste, Saloff-Coste, 2011). 2011). 30 function such that . This can be interpreted as 20 20 solutions solutions (in ) (in of heat ) of equations. heat equations. 17 examples are given in (A 20 solutions (in ) of heat equations. 23 23 is is a a nonnegative nonnegative function function such such tht 22 Definition heat of asetstrongly local regular Dirichlet form on 16 kernel for , where is definition any bounded interval, and 193. An associated in the above yields the definition of loca 22 Definition 3.ofof An associated hea ________________________________________________________ 22open Definition 3. An associated heat kernel of aclosure strongly local regular Di 19 insetthe abo but a31generalization ofthat thefixing classical equation 21 21 1818 Note Note fixing an open setwith and and replacing replacing with with the the closure the the set 21 ǡ that ሺ 14 ǡ an ሻheat for all test functions . However, what are test functions? 14 14 14 18 Note that fixing 2 2 heat 24 24 ofgiven ofisthe the heat equation equation associated associated toto satisfies satisfies 1715 examples are in (Aronson, 1968), see also (Gyrya Saloff-Coste, 201 23Simple examples is20a nonnegative function such that any solution (in of heat equations. ofsolutions solutions in the sense introduced above are functions 23 is a& nonnegative functi all 23 athe nonnegative function such that aa This means This means all .of 20 solutions (in )strongly of equations. 22 22 Definition Definition 3. An 3. associated An heat kernel heat kernel of anonnegative of aheat strongly local local regular regular Dirichlet Dirichlet form for Simple examples of weak solutions in the sense introduced above 1919 inassociated in the above above definition definition yields yields the the definition definition ofsense local local the Laplacian ∆weak . Informally, a )solution of 15 the heat 22 Definition 3.for An associated heat kernel of 15 Simple examples of weak solutions in the sense int 32 Given a Hilbert space and an open interval , denote the Hilbert space 15 Simple Simple examples examples of of weak weak solutions solutions in in the the sense introdu intro 21 19 21 25 25 and and . . 14 18 Note that fixing an open set and replacing with the 24 of20thesolutions heat equation associated to satisfies for all for 20 , where is any bounded interval, and . More interesting 24 of the heat equation associated 24 of the heat equation associated to satisfies 23 equations. is a nonnegative is26 a nonnegative function function such that such any that nonnegative any interva nonneg 16equations. for ,strongly where local is, where any bounded interval, and solutions (in(in ) 23 of ) ofheat heat 23 is a20 nonnegative function 16 kernel for , where is any bounded 33 all measurable functions with finite 26 22 of examples Definition An associated of a norm regular form o 16 16 for , associated where isDirichlet is any any bounded bounded solutions (in )ainterval, of heatan eq 22 for Definition 3. An heat kernel ofinterval, strongly 15 Simple of3. weak solutions inheat the sense introduced above are functions 19 in the above definition yields th 25 are . amples givenand in (Aronson, 1968), see also (Gyrya &17 Saloff-Coste, 2011). 2121 25 and . 25 and . 17 examples are given in (Aronson, 1968), see also (Gyrya & Saloff-Coste, 20 24 24 of the of heat the equation heat equation associated associated to to satisfies satisfies examples are given in (Aronson, 1968), see also (Gyrya & 21 24 of the heat equation associated to 27 27 The The author author ends ends this this section section with with the the definition definition of of the th 23 is aนติ20 nonnegative function such that any nonnegative solution 17 17 examples examples are given given in (Aronson, (Aronson, 1968), 1968), see see also also (Gyrya (Gyrya && Salo Sa 26Note2222 สัwhere ทาเสนา /replacing วารสารวิ ทยาศาสตร์ บheat ูรพา. 19are (2557) 1inregular : a169-177 23 is nonnegative function 171 26 26 16 for ,set isheat any bounded interval, and . More interesting solutions equations. Definition 3.An An associated heat kernel kernel of of aof astrongly strongly local local regular Dirichlet form form on on that Definition fixing an 3.open with the closure ofDirichlet the set 18 and Note fixing an open set and replacing with th 25associated 2521 and and .(in18 .)that 22 Definition 3. An associa Note that fixing an open set and replacing 28 28 inequality. inequality. 25 and . 18 18 Note Note that that fixing fixing an an open open set set and and replacing replacing 27 The author ends this section with the definition of the (uniform) parabolic Harnack 24 of the heat equation associated to satisfies for 27 ends this(unas 27 The ends this section with theThe definition of the 17 examples are given innonnegative (Aronson, 1968), see also (Gyrya &ofSaloff-Coste, 2011). 24author the heat equation associated to author satisfies 26 26 19in 2323 isisa anonnegative function function such such that that any any nonnegative nonnegative solution solution the above definition yields the definition of local 26 29 29 in the above definition yields is aabove nonnegative 19 in the above det 22open Definition 3.ends An associated heat kernel of a 23 strongly local regular Dirichle 28 inequality. 1919 in in the the above definit defin 25the and . 28 28 inequality. inequality. 18 Note that fixing an set and replacing with the closure the set 25 and . 27 27 The author The author ends this section this section with with the definition the definition of the of (uniform) the (uniform) paraboli pa lutions (in24 heat equations. 24) ofof of the heat heat equation equation associated associated to to satisfies satisfies for for all all 27 The author ends this section with 30 30 Definition Definition 4. 4. A A strongly strongly local local regular regular Dirichlet Dirichlet form form ass as solutions (in ) of heat equations.whose distributional derivative can Let 34 be20the set of all functions 29 7 will be viewed asdenote a function ions assatisfying aInfunction from a.e. to on , 4any function theasisset all and functions such that for .any relative also order of heat a function to , Denote any function efore, a ofwell-defined object. 44 hat , the -dualfrom ctionDirichlet . so a solution 4. weak ular formto view .Note Aequations isofaso 11on 5 . subset Note that that of so(local) the -dual -dual 8function 4,, the . , there exists a function satisfying a the set of all functions such that for any relatively compact open will be viewed as a function . 4 and also denote s athe function from-dual to of ,221any function Note soan ,well-defined thea heat -dual of is that aDefinition well-defined object. Denote of heat equation if .Therefore, 9 2. Let be open time interval and be operato , the Therefore, is is a a well-defined objec obje to view a solution heat as a function from to of. the,and ,and there exists a function satisfying a.e. on ofcompact . equations e interval be a heat operator 6associated toIn aorder strongly set of all functions such any relatively open 4 afor and also denote 2 time Therefore, is, , athe well-defined object.als 10 and localthat regular Dirichlet form on function ndDefinition be a heat operator associated to ainterval strongly a well-defined object. Denote 3 3 and and als 2. Let be an open be heat operator associated to a strongly 1 Note that so the -dual of 1 Note that so of . .D 7 will be viewed as a function . . A-dual view a. A solution ofcompact equations as functionweak from , a.e. any on function on tointerval function is a (local) eset that soheat , relatively the , to-dual of . et of , there exists a function satisfying . yorder open relatively in and any 1 Note that so , the -dual of . of all functions such that for any compact open solution of the heat equation is aisis .A function weak 3 isa astrongly and Denote alsocom 8(local) and denote 4to 4 the the set set of offunction all all functions functions such such that that for for any any relatively relatively com regular Dirichlet form on11 .A (local) weak ,elocal the -dual of associated . Therefore, Therefore, well-defined object. Denote 22function aaif well-defined object. aIn heat will beoperator viewed asaalso asolution if is.as aa2.well-defined object. Denote order to view of heat equations function from to atime ,interval any function , there exists a function satisfying a.e. on . 12 a) , and 2 Therefore, is well-defined object. Denote 9 Definition Let be an open and be a heat operator associa 4 the set of all functions such that for any relatively compac for any relatively compact information about Harnack inequality including its variance 5open 5(local) subset subset of of ,,there exists existsaafunction function satisfying satisfying a.e. on on solution ofobject. the heatDenote equation if ell-defined function weak and also a.e. denote 33 asisaafunction and also denote soview , the of from .there be viewed .Dirichlet and also denote to awill solution of heat equations as local a-dual function to , any function 13 b) for any open interval relatively compact in and any 5 subset of , there exists a function satisfying a.e. on 10and regular form on . A function 3 and also denote isfying a.e. on . 6 6 In In order order to to view view a a solution solution of of heat heat equations equations as as a a function function from from to to , , an an 2. Let be an open time interval be a heat operator associated to a strongly a) , and and history, see (Kassman, 2007). 4 a well-defined the setthat of all all functions such that that for any any relatively relatively compact compact open open and also 4is set of functions such for ompact in and any denote , the object. Denote thebeany set of all functions such for any relatively compact open ill viewed as a function . and , 6 In order to view a solution of heat equations as a function from to , any fu 11 solution of the heat equation if 4 the set of all functions such that for any relatively compact open tion from to any function 7on 7 interval will be viewed viewed as asaafunction function .. lar Dirichlet form A function is a, (local) weak b)relatively for open interval relatively compact in, ,will and any nition 2.any be an, open time be abe heat operator associated subset of. and there exists function satisfying a.e.on on ny compact 55function subset of there exists aafunction satisfying a.e. .. of examples ,Let there exists aopen satisfying a.e. on . to a strongly and also denote 8 8 12 a) , and 7 will be viewed as a function . imple of weak solutions in the sense introduced above are functions 5 subset of , there exists a function satisfying a.e. on . the heat equation if regular Dirichlet form on . A function is a (local) weak , 6 In order to view a solution of heat equations as a function from to , any function gny a.e. on . 6 In order to view a solution of heat equations as a function from to , any function et be view an open timethat interval andrelatively beas a aheat operator associated ainterval strongly rder to a solution of for heat equations function from ,to function such compact open 8view 9any 9 Definition Definition 2.2.any Let Letopen be beinterval an antoopen open time time interval and and be be a,aany heat heat operator operator associated associated to to Laplacian 13 b) for compact in anyfunction , rion , where istoany bounded interval, and .any More interesting 6, and In order a solution of heat equations as arelatively function from 14 of the heat equation if om to , any function 1 aand 7 will be viewed as a function 7 will be viewed as a function .to. .be irichlet form on . A function is a (local) weak 9 Definition 2. Let be an open time interval and heat operator associated to a st will be viewed as a function . unction satisfying a.e. on . operator associated to a strongly 10 10 local local regular regular Dirichlet Dirichlet form form on on . A A function function is is a a (lo (l Simple examples of weak solutions in the sense sopen are given alsoand (Gyrya & Saloff-Coste, 7 in (Aronson, will besee viewed as aany function A. classic example heat equations isAthe classical 15 Simple examples of weakofsolutions in the sense introduced abo 881968), interval relatively compact in , 2011). ,sense and 2 function classic example of h heat equation if in the introduced above are functions 10 local regular Dirichlet form on . A is a (local fnlutions heat equations as a function from to , any function 8introduced is a open (local) weak 11 11 solution solution of of the the heat heat equation equation ifif ote that fixing an set and replacing with the closure of the set 9 Definition 2. Let be an open time interval and be a heat operator associated to a strongly 9 Definition 2. Let be an open time interval and be a heat operator associated to a strongly introduced above are functions for 16 for , where is any bounded interval, and 1 Laplacian e2.any sense above are functions r open interval relatively compact in and any , Simple examples of weak solutions in the sense introduced above are functions Let open time interval be. of aMore heat operator associated to aifoperator stronglyassociated to a strongly , and stor any bounded interesting 11 beand solution thedefinition heat ction .10 9 be an Definition 2.and Let an open time interval and be aequation heat heat with the Laplacian on 12 12 a) a) the ,equation ,and and 3 is see asweak the h in above yields the definition of local associated tointerval, a strongly 10 local regular Dirichlet form on aon (local) weak local regular Dirichlet form on . .weak AA function isisa& (local) 2.interesting A classic example of heat equations the also classical heat equation with 17 examples are given in (Aronson, 1968), (Gyrya Saloff-Cost ded interval, and . More 14 , where I is any bounded interval, and for , where is any bounded interval, and .function More interesting ar Dirichlet form on A function is a (local) se introduced above areSaloff-Coste, functions interval relatively compact in and any , 8), see also (Gyrya & 2011). 12 a) , and 10 local regular Dirichlet form on . A function is a (local) weak 13 13 1 b) b) for any open open interval interval relatively relatively compact compact in in and and any anymeasure. ,, above Laplacian (in ) ofequation heat equations. 11(Aronson, solution theany heat equation isSaloff-Coste, a (local) weak 11 solution ofoffor the heat equation ifif 18 Note that fixing an open and introduced replacing with (Gyrya &and 15 weak solutions in the fu 4set Itsmeasure associated 1 examples Laplacian examples are given in 1968), see also (Gyrya & Saloff-Coste, the heat if interval be a of heat operator associated to aSimple strongly asof the heat2011). operator and the Lebesgue More interesting are given inset terval, and ..2011). More interesting 3 examples on heat operator and thesense Lebesgue measure asDirichle itsare und set and replacing with the closure of the 13 b) for any open interval relatively compact in and any , 11 solution the heat equation if , 2 A classic example of heat equations is the classical heat equation with the 12 a) , and 12 a) , and 19 in the above definition yie replacing with the closure of the set 16 for , where is any bounded interval, and 2 above are A classic example of heat equations mple of weak solutions ina the introduced functions Note that fixing analso open set&sense and replacing with the closure of the set 5is the classical hea. , and onexamples .above A function is a local (local) weak an 3. & Saloff-Coste, 2011). in the(Aronson, definition yields the definition of local 1968), see (Gyrya Saloff-Coste, 2011). An associated heat kernel of strongly regular Dirichlet form on 12 a) , and 4 measure. Its associated Dirichlet form is defined by 13compact b) for any open interval relatively compact inoperator and any 13 b) for any open interval relatively compact in and any ,, & 20 solutions (in ) of heat equations. e above definition yields definition ofinterval, local 17 examples are given in (Aronson, 1968), see alsoLebesgue (Gyrya Saloff-Coste, 2011). 1the Laplacian asthe its underlying volume measure. Its associated Dirichletits underlyin 3bounded on as heat and the measure , where isthe any and . yields More interesting insense the above definition definition local open interval relatively in and any , and Simple of weak solutions in the introduced above are functions cing the closure of set 3inthat onthe the heatofoperator and theasLebesgue me is , aif nonnegative such any nonnegative solution 13 examples b)with forfunction any open interval relatively compact any , as is 21 V and replacing Note that fixing an open set 14 14 18 Note that fixing an open set and replacing with the clo 2 A classic example of heat equations the classical heat equation with the are given in(in (Aronson, 1968), see also (Gyrya Its &interval, Saloff-Coste, 2011). solutions ) ofsolutions heat equations. form is defined by for , where is any bounded and . More interesting ve definition yields the definition of local 4 measure. associated Dirichlet form is defined by examples of weak in the sense introduced above are functions at equation associated to satisfies for all 22 Definition 3. An associated heat kernel of a strongly local regular Dir 1 Laplacian 14 15 15 Simple Simple examples examples of of weak weak solutions solutions in in the the sense sense introduced introduced above above are are function functio 4 measure. Its associated Dirichlet form is defined by 1with Laplacian D(E) with the regular closure ofDirichlet setreplacing has 19 in5 the with abovethedefinition yields the d rnel ofareainfixing strongly local form on as te that an open set1968), and thetheclosure of theandset domain and any ,the mples in (Aronson, also (Gyrya & is Saloff-Coste, 2011). 3see on heat operator the Lebesgue measure as itsclassical , where is any bounded interval, and . equations. More interesting dompact . given 23 is awhere nonnegative function such that an ongly regular Dirichlet form on 15 Simple examples of weak solutions in the sense introduced aboveisare functions 2local Aassociated classic example of heat equations the classical heat equation with the Laplacian uced above are functions 16 16 for for , , where is is any any bounded bounded interval, interval, and and .underlyi . More More Definition 3. An heat kernel of a strongly local regular Dirichlet form on 2 A classic example of heat equations the 20 solutions (in ) of heat 14 compact support inthat in1the above definition yieldsyields 14 in the above definition the the definition ofLaplacian local such any nonnegative solution 6 set Dirichlet norm is nothing but th Note that fixing an open set and replacing with closure of the ven in (Aronson, 1968), see also (Gyrya & Saloff-Coste, 2011). such anyform nonnegative 16 examples for , where isinin bounded interval, and . space More of inteo 24 ofassociated the heat equation associated to(Gyrya satisfies 21 17 17 are aregiven given in in (Aronson, (Aronson, 1968), 1968), see see also also (Gyrya & Saloff-Coste, Saloff-Coste, 2011). 2011). 14heat nd . that More interesting 4examples measure. Its Dirichlet form issense defined by solution isends aequations. nonnegative function such that any nonnegative 5solution with the domain ,& the local Dirichlet on 15of Simple examples of weak solutions the introduced above areSobolev functions 15 Simple weak the sense introduced above are functions in ) of 3regular on the heat operator and Lebesgue measure as its underlying volume satisfies for allsolutions he author this set section with definition ofof the (uniform) parabolic Harnack 2the Agiven classic example of heat equations isas the classical heat equation with the in the above definition yields the definition of local Vexamples ) of heat equations. the definition local solutions (in ple examples of weak solutions in as the sense introduced above are functions 3the on the heat operator and the Lebesgue at fixing an open and replacing with the closure of the set 22 Definition 3. An associated heat kernel of a strongly local regular Dirichlet 17 examples are in (Aronson, 1968), see also (Gyrya & Saloff-Coste, 2011). tisfies for all 25 and . 15 Simple examples of weak solutions in the sense introduced above are functions off-Coste, 2011). 18 18 solution Note Note that that fixing fixing an an open open set set Sobolev and and norm replacing replacing with with the the closure closurefo of the heat equation associated to satisfies Dirichlet but the 1 is nothing such anyequations. nonnegative bove are functions 16 for 6 interval, , where where is any any bounded interval, and 5for all Laplacian More interesting 16 for ,norm is bounded interval, and . . More interesting y. ions (in )that of heat , where is any bounded and . More interesting 26 4 measure. Its associated Dirichlet form is defined by Definition 3. An associated heat kernel of a strongly in the above definition yields the definition of local 5 a23 with thelocal domain ,above the Sobolev space ofdefinitio order is abounded function such any nonne 18of Note that fixing anmeasure. set and replacing the closure of ht Its associated form iswith defined by 3strongly on asopen the heat and the Lebesgue measure as its underly 16and , for where is any interval, andoperator . More interesting with the closure the set 19 19 in in the the above definition definition yields yields the the definiti 3. An associated heat kernel regular form on .for 24 Dirichlet A classic example of heat equations is the classical with the domain the with the domain 5nonnegative ,,that the 17of examples arefunctions given (Aronson, 1968), see also (Gyrya &Dirichlet 2011). 17 examples are given inin (Aronson, 1968), see also (Gyrya & Saloff-Coste, all . the More interesting 5Saloff-Coste, 1& Laplacian 1 Saloff-Coste, Laplacian 27 The author ends this section with the2011). definition of theSobo (uni re given in (Aronson, 1968), see also (Gyrya 2011). utions in sense introduced above are 2 is nothing of heat equations. 6 Dirichlet norm but the Sobolev norm 19 in the above definition yields the definition (E,D(E)) on L (X, ν ) is a local regular Dirichlet form 24 of the heat equation associated to satisfies 17 examples are given in (Aronson, 1968), see also (Gyrya & Saloff-Coste, 2011). yields the definition of20 local 20 solutions solutions (in (in )such )local of of heat heat equations. equations. snition function that anyDirichlet solution ntion the definition ofheat the (uniform) parabolic Harnack 2nonnegative nawith 4.nonnegative A strongly local regular Dirichlet form associated to a heat operator on 3. An associated kernel of a strongly regular Dirichlet form on 7 and the intrinsic distance is noth 6 Dirichlet norm is nothing but the Sobolev norm ste, 2011). 18 Note that fixing an open set and replacing with the closure of the setom 4 measure. Its associated form is defined by 18 Note that fixing an open set and replacing with the closure of the set L -Sobolev space of order one. The Dirichlet norm is 2 A classic example of heat equations is the classical heat equation 2 A classic example of heat equations is the classical heat equation 28 inequality. of the (uniform) parabolic Harnack thatbounded fixing an open set and replacing with the closure of the set sedefinition any interval, and . More interesting 3 on as the heat operator and the Lebesgue Laplacian 21 21 The author ends this section with the definition of the (uniform) parabolic Harnack 20 (in )domain heat equations. 25 and .exists Note fixingsolutions aninequality open set and replacing with the of Sobolev the tth equation associated to for allclosure satisfies (uniform) parabolic Harnack ifofthere aany constant such nonnegative function such is 18 aclosure nonnegative function such that nonnegative solution 8, the yields The heat kernelof 5 satisfies with the space of order 29 19 in the the above definition yields theset definition of local the of kernel the19 setthat in above definition the definition local An associated heat ofof a22 strongly local regular Dirichlet form on in the above definition yields the definition of local 8), see also (Gyrya & Saloff-Coste, 2011). 21 nothing but the Sobolev norm A classic example heat equations is the classical heat equation with the Laplacian 26 22 Definition Definition 3. 3. An An associated associated heat heat kernel kernel of of a a strongly strongly local local regular regular Dirichlet Dirichlet form form inequality. 7 . and the intrinsic distance is nothing but the Euclidean distance on 3 on as the heat operator and the Lebesgue measure asasits ition of the (uniform) parabolic Harnack 3 on as the heat operator and the Lebesgue measure it 4 measure. Its associated Dirichlet form is defined by in the above definition yields the definition of local 19 Laplacian . any , and any non-negative weak solution of the heat equation 6 Dirichlet norm is nothing but the Sobolev norm 30 Definition 4. A strongly local regular Dirichlet form associa e heat equation associated to satisfies for all 20 solutions solutions (inuclosure of heat equations. the definition local 20 (in of heat equations. of))heat the heat equation that anyofnonnegative solution nnegative function such that any nonnegative solution 22 Definition 3. An associated heat kernel of a strongly local regular Dirichlet form ) of heat equations. 27 The author ends this section with the definition of the (uniform) pa set and replacing with the of the set gular Dirichlet form on 23 23 is is a a nonnegative nonnegative function function such such that that any any nonnegative nonnegative rnelds Dirichlet form associated to a operator on with the 520 example ,Its the Sobolev space of order one. Theto is ,defined 8 the 431 The heat kernel associated (in of heat equations. 21 21 Aand classic ofdomain heat equations is classical heat equation with the Laplacian 5Its withassociated the domain So on as)to the heat operator and the Lebesgue measure as its underlying volume ,A satisfies (uniform) Harnack inequality ifthe there by exis orm associated ayields heat operator on .solutions measure. associated Dirichlet form isis defined by 4intrinsic measure. associated Dirichlet form defined byon 4. strongly local regular Dirichlet form toparabolic aEuclidean heat operator associated to (E,D(E)) satisfies associated to satisfies for all 23 is a nonnegative function such that any nonnegative solu 28 inequality. in above definition the definition of local 7 . and the distance is nothing but the distance on 21 that anythe nonnegative solution chDefinition Harnack inequality if there exists a constant such eation author ends this section with the definition of the (uniform) parabolic Harnack 24 24 of of the the heat heat equation equation associated associated to to satisfies satisfies 6 Dirichlet norm is nothing but the Sobolev norm 22 Definition 3. An associated heat kernel of a strongly local regular Dirichlet form on 22 Definition 3. An associated heat kernel of a strongly local regular Dirichlet form on 7 and the intrinsic distance is nothing but the Euclidean distance 6 Dirichlet norm is nothing but the Sobolev norm 32 that for any , and any non-negative weak solution of inequality if there exists a constant such 3..measure. An 22 associated kernel of a 8form strongly local regular form on 29 called, forofthe res 9form satisfies (uniform) parabolic Harnack inequality ifasthere exists a constant such 5and with the domain , thewhich Sobolev spaceby order ItsDefinition associated is heat defined by on asheat the heat operator the Lebesgue measure itstounderlying volume associated to aDirichlet heat operator on 24 of the heat equation associated satisfies 3. An associated kernel aDirichlet strongly local regular Dirichlet on The heat kernel associated towill isbe defined non-negative weak solution ofwith the heat equation for all and for all 25 25 and and .of ..function The author ends this section the definition of the (uniform) parabolic Harnack Dirichlet form on 23 is a nonnegative function such that any nonnegative solution 23 is a nonnegative such that any nonnegative solution 8 The heat kernel associated to 33 such on , the ve weak solution of heat 30 Definition 4. strongly local regular Dirichlet to a h athat nonnegative function nonnegative solution 10 form solution Denoteassociated the Gaussia for any , 26 and equation any non-negative weak solution of heat equation 6 function Dirichlet norm isAany nothing but the Sobolev norm 26 ality ifassociated there exists a is constant such 23 athe nonnegative such that any nonnegative 25 and . that uality. e. Its Dirichlet form is defined by The author ends this section with the definition of any nonnegative solution 7 . and the intrinsic distance is nothing but the Euclidean distance on 24 of the heat equation associated to satisfies for all 24 of the heat equation associated to satisfies for 9 which will be called, for the rest of this work, the classical heat kernel. with the domain , and both 5 supremum and infimum are , the Sob hor ends this section with the definition of the (uniform) parabolic Harnack 31 form satisfies parabolic Harnack inequality ifEuclidean there existsparaboli a cons nel of a strongly regular form on (uniform) equation to satisfies for all 26Dirichlet mean and variance .all Th 4. solution A strongly regular Dirichlet associated a heat operator on 27 27 The The author author ends ends this this section section with with the the definition definition of ofzero the the (uniform) (uniform) parabol on , equation andto the intrinsic distance is11nothing but the ak of local the heat 24 associated of local the heat equation associated to satisfies for all 8 The heat kernel associated to is defined by for all 25 and . 25 and . 10 Denote the Gaussian kernel i.e. the density of the Gaussian(Normal) 6 Dirichlet norm is nothing but the Sobolev norm the (uniform) parabolic Harnack inequality. 32 that for any , and any non-negative weak solution of the hea 27 The author ends this section with the definition of the (uniform) parabolic Hd such that any nonnegative solution . dnition up (uniform) to725 sets of measure zero. 28 Dirichlet inequality. inequality. the parabolic Harnack 12 a random variable. Actually, atisfies (uniform) parabolic Harnack inequality if there a rest constant such 5will the domain , , the Sobolev space 5but with with theexists domain the Sobolev space 4. Aand strongly local regular form associated tothis a heat operator on distance on . . the intrinsic distance is nothing the Euclidean distance on and .28 26 26 9 which be called, for the of work, the classical heat kernel. 7 and the intrinsic distance is nothing but the Euclidean distance mean zero and variance .beThen is both aheat proba 29 29 which will called, for restnorm of this work, the classical heat ker 9ifform 34 where , the 13 and su 28 Harnack inequality. 3311 regular ,nothing Definition 4. A strongly local Dirichlet satisfies for allsection In i.e. general, any op nywith , and any non-negative weak solution of the heat equation 6inequality Dirichlet norm isis but the Sobolev 6on Dirichlet norm nothing but the Sobolev norm satisfies (uniform) parabolic there exists ato constant such 27 The author ends this section with the definition of the the (uniform) parabolic Harnack the domain ,A the Sobolev space of order one. The 27 The author ends this with the of (uniform) parabolic Harnack 826 The heat kernel associated isdefinition defined by and both supremum and infimum are 10 Denote the Gaussian kernel i.e. the density of the Gaussian(Normal) distrib strongly local regular Dirichlet form associated to a heat operator on The heat kernel 8 The heat kernel associated to author ends this section with the definition of the (uniform) parabolic Harnack 29 nformally, a heat operator satisfies parabolic Harnack inequality if its solutions, further 30 30 Definition Definition 4. 4. A strongly strongly local local regular regular Dirichlet Dirichlet form form associated associated to to a a heat heat ope op 12 a35 random variable. Actually, isbut the transition density of the Brownian o 2 the 10 Denote the Gaussian kernel i.e. theis density ofmotion the Gaus and supremum and are 7 infimum . transition and the isof nothing the Euclidean distance on 27parabolic The author ends this section with definition ofof the (uniform) parabolic Harnack heat kernel the den , both where ,Sobolev and both supremum and14infimum are computed up to sets measure zero. (E,D(E)) associated to a heat operator Lintrinsic on Lvariance (X,distance νa)local for any ,28 and any non-negative weak solution the heat equation Dirichlet norm is nothing but the norm 28 inequality. inequality. niform) Harnack 11 mean zero and . Then i.e. is probability es (uniform) parabolic Harnack inequality if there exists constant such 30 Definition 4. A strongly regular Dirichlet form associated to a heat operato massociated the boundary, are roughly constant. One consequence of uniform parabolic Harnack associated to ∆ is defined by to a heat operator on 31 31 satisfies satisfies (uniform) (uniform) parabolic parabolic Harnack Harnack inequality inequality if if there there exists exists a constant constant he domain , the Sobolev space of order one. The 13 In general, any heat operator is a Markov operator of some Markov pr 9 which will be called, for the rest of this work, the classical heat kernel. 36 11 mean zero and variance . Then i.e. 8 The heat kernel is defined bypossibl 28 both inequality. 15 associated 2012). toTherefore, it is 29 measure 29 ,(uniform) and supremum and infimum aresolution computed up toany sets of zero. satisfies parabolic Harnack inequality ifof 12norm acontinuous random variable. is the transition of the motion ondistrib ,adefinition and non-negative weak the 7Actually, theequation intrinsic distance is16 nothing butBrownian the Euclidean distance o. 31 satisfies (uniform) parabolic Harnack inequality ifof there exists aboth constant nynet with of the (uniform) parabolic Harnack is that solutions always admit representative so one may assume that norm issuch nothing but the Sobolev 29the ere exists constant such 32 32 that that for for any any ,and ,heat and and any any non-negative non-negative weak weak solution solution of of the the heat heat equa equa 14 heat kernel isthere the transition of the corresponded Markov Processes (Chen 37 Informally, adensity heat operator satisfies parabolic Harnack inequal 10 Denote the Gaussian kernel i.e. thedensity density the Gaussian(Normal) 12 a random variable. Actually, is the transition density of the Brow 34 where , and supremum 30 Definition 4. A strongly local regular Dirichlet form associated to a heat operator on subject will not be discussed he 30 Definition 4. A strongly local regular Dirichlet form associated to a heat operator on 4. A strongly local regular Dirichlet form associated to a heat operator on atisfies parabolic Harnack inequality if its solutions, further 9 which will be called, for the rest of this work, the classical heat kernel. 13 In general, any heat operator is a Markov operator of some Markov processe , exists a constant H > 1 such that for any 8 The heat kernel associated to 32 that for any , and any non-negative weak solution of the heat equation continuous. For further information about Harnack inequality including its variance and 30 4. local regular Dirichlet associated to aoperator heat operator on which will be called, for the rest of isthis work, the classical heat 9form on ofa Informally, theDefinition heat equation 15 2012). Therefore, it is possible to study Markov processes via heat 33 33 solutions, on on ,,supremum away from the are constant. One consequence of u 0strongly bolic inequality its further 11 mean zero and variance .boundary, Then i.e. is a equations. probability 13 In general, heat aaOne Markov operator of so 738 . . such and the intrinsic distance is nothing but the Euclidean distance on 7(uniform) and the intrinsic distance is nothing but the Euclidean distance on a,31 heat operator satisfies Harnack inequality ifany itsroughly solutions, further possible question ated toHarnack heat operator onAif 31 satisfies (uniform) parabolic Harnack inequality if17 there exists constant such satisfies parabolic Harnack inequality if there exists constant and both and infimum are 35 computed up to sets ofof measure zero. ytisfies constant. One consequence of uniform parabolic Harnack (uniform) parabolic Harnack inequality ifparabolic there exists a constant such 14 heat kernel is the transition density of the corresponded Markov Processes (Chen & F 10 Denote the Gaussian kernel i.e. the density the Gaussian(Normal) distribution with 33 on , ee (Kassman, 2007). 31 satisfies (uniform) parabolic Harnack inequality if there exists a constant such 10 Denote the Gaussian kernel i.e. the density of the G 16 subject will not be discussed here. 39 inequality is that such solutions always admit continuous representativ One ofif uniform Harnack and any non-negative weak solution u of the heat 12 awhich random variable. Actually, is uniform thethis density of Brownian on to btP heat kernel transition density of the corresponded 36 8may The heat kernel associated to isisMarkov defined 8operator The heat kernel associated tomotion defined away from the boundary, are roughly constant. One consequence of parabolic Harnack 18 operators behaves similarly 32 thatparabolic for any and any non-negative weak solution of the heat equation equation ists a consequence constant such 32 for any , ,14 and any non-negative weak solution of the heat will be called, for the rest work, the classical heat kernel. 9but .equation and the intrinsic distance is nothing Euclidean distance on inequality itsnon-negative solutions, further eDirichlet ,that and both supremum and infimum are ays admit continuous representative so one assume , and any weak solution of the heat ryHarnack form associated to athe heat on which will beistransition called, for the rest ofthe this work, the up to sets of measure zero. 15 2012). Therefore, itthat is possible toof study Markov processes via heat equations. How 11 mean zero and variance . Then i.e. is a probability density of 32 that for any , and any non-negative weak solution of the heat equation 17 One possible question that whether or not heat kernels associated 40 they are continuous. For further information about Harnack inequality 11 mean zero and variance . Then i.eh ontinuous representative so one may assume that 37 Informally, a heat operator satisfies parabolic Harnack inequality its 13 In general, any heat operator is a Markov operator of some Markov process 15 2012). Therefore, it is possible to study Markov processes via inequality isof that such solutions always admit continuous representative so i.e. onethe may thatGaussian(Normal) ifdistri 19assume equation on of the heat equation 33 on 10 33 on , ,, be Denote the Gaussian kernel density of the The heat kernel associated to is defined by consequence uniform parabolic Harnack rmation about Harnack inequality including its variance and , c Harnack inequality if there exists a constant such 16 subject will not discussed here. puted up to sets of measure zero. classical heat kernel. , and both supremum and infimum are . eout intrinsic distance is nothing but the Euclidean distance on 12 a random variable. Actually, is the transition density of the Brownian motion on . 33 on , 18 operators behaves similarly to the Gaussian kernel. 41 history, see (Kassman, 2007). which will be called, for the rest of this work, the classical heat 9 Harnack inequality including its variance and where where 34 34 , , and and both both supremum supremum and and in 12 ainequality random variable. Actually, is Markov the transition density of the B 38 that away from boundary, are roughly constant. One consequence p 14 heat kernel is the the transition density of the corresponded Processes (Chen &in Fk 16 subject not be discussed here. they areacontinuous. Forsatisfies further information about Harnack including its20 variance and ormally, heat operator parabolic Harnack inequality if.will its solutions, further 11 mean zero and variance Then i.e. isofa uniform probability ous representative so one may assume non-negative weak solution of the heat equation 17 One possible question is that whether or not heat kernels associated to 13 In general, any heat operator is a Markov operator of some Markov processes and the The heat kernel associated to is defined by 34 are15 where , One and both and infimu 19One 42 called, Denote the Gaussian kernel i.e. the density 13of In general, any heat operator is asupremum Markov operator ofo both supremum and infimum 10 Denote the Gaussian kernel i.e. the density of the Ga sets of measure zero. 39 inequality is17that such solutions always admit continuous representative so one 2012). itmeasure iscalled, possible to study Markov processes via heat How possible question is that whether orequations. not heat ker history, see (Kassman, 2007). 21 In the previous section mrnack the boundary, are roughly constant. uniform parabolic 35 35 computed computed upTherefore, to to sets sets of of measure zero. zero. 9consequence which will be for the rest of this work, the classical heat kernel. 9up which will be for the restHarnack of this work, thethe classical heat kernel. Informally, a heat operator satisfies parabolic Harnack inequality if its solutions, further inequality including its variance and 12 a random variable. Actually, is the transition density of Brownian motion on 18 operators behaves similarly to the Gaussian kernel. 14 heat kernel isalways the transition density ofare the corresponded Markov Processes (Chen &Behavior Fukushima, 43 20 Gaussian 36 36 14 heat kernel is the density ofmean the corresponded Marko 40continuous they continuous. For further information Harnack 11 mean zero and variance .about Then i.e. 16 subject will not be discussed here. 18of operators behaves similarly to the Gaussian kernel. 35 computed up sets measure zero. ofheat Gaussian (Normal) distribution with zero and isfrom that such solutions admit representative sothe one may assume that 22 intrinsic distance i.e. including theproces Eucl 34 the ,any and both supremum and infimum are where 34 where ,heat and both supremum and infimum are 10 Denote the Gaussian kernel i.e. the density of the Gaussian(Norm 10 Denote the Gaussian kernel i.e. the density ofinequality the Gaussian(Norm ylly, the boundary, are roughly constant. One consequence of uniform parabolic Harnack 13 Into general, operator is equations. atransition Markov operator of some Markov which will be called, for rest of this work, the classical kernel. , and both supremum and infimum are 19 a15 heat operator satisfies parabolic Harnack inequality if its solutions, further 2012). Therefore, it is possible to study Markov processes via heat However, that 36 where 37 37 Informally, Informally, a a heat heat operator operator satisfies satisfies parabolic parabolic Harnack Harnack inequality inequality if if its its solutio solutio 21 In the previous section, the classical heat kernel depends on two obje 34 , and both supremum and infimum are where 15 2012). Therefore, it is possible to study Markov processes vpF 41 history, see (Kassman, 2007). 17 One possible question is that whether or not heat kernels associated to 12 a random variable. Actually, is the transition density of the 19 ontinuous. For further information Harnack inequality including its variance and 23 that underlying space It isisa& in supremum and infimum are uality is that such solutions always admit continuous representative so one may assume 14about heat kernel isofthe transition density the corresponded Markov Processes (Chen 11 mean zero and variance .of i.e. pr variance .. Then Then i.e. 11 mean zero and variance Then i.e. aBr 35 computed up to sets of measure zero. 35 computed up to sets of measure zero. Denote the Gaussian i.e. the density of the Gaussian(Normal) distribution with boundary, are roughly constant. One consequence uniform parabolic Harnack 20kernel Gaussian Behavior 16called, subject will not be discussed here. 37work, Informally, a heat operator satisfies parabolic Harnack inequality if its solutions, up to sets of measure zero. 38 38 away away from from the the boundary, boundary, are are roughly roughly constant. constant. One One consequence consequence of of uniform uniform paraboli parabol inequality if its solutions, further 22 intrinsic distance i.e. the Euclidean distance on , and another is the dime will be for the rest of this the classical heat kernel. 16 subject will not be discussed here. 42 In any heat is via aConsider Markov of 18 operators similarly toActually, thegeneral, Gaussian kernel. 2013inequality Gaussian Behavior eare (Kassman, 2007). 35 computed up of measure zero. 24 operator kernels. aoperator heat opera 36 toboth 36 continuous. For information about Harnack including its and 15 2012). Therefore, iti.e. is to study Markov processes heat equations. Ho 12 aabehaves random variable. is the transition density of the Brownian motio 12 random variable. Actually, isvariance the transition density of the Brownian mot and both supremum and infimum are iskernels probability density ofwhether arepresentative random variable. mean zero and variance .sets Then is aaprobability density of and supremum and infimum are 21 In the previous section, the classical heat kernel depends on two objects, at such solutions always admit continuous representative so one may assume that 1736 Onefurther possible question is that whether or notpossible heat associated to other heat 38 away from the boundary, are roughly constant. One consequence of uniform parabolic Haao 39 39 inequality inequality is is that that such such solutions solutions always always admit admit continuous continuous representative so so one one may may nce of uniform parabolic Harnack 23 underlying space It is interesting to know this behavior happens 17 One possible question is that whether or not heat Denote the Gaussian kernel i.e. the density of the Gaussian(Normal) distribution with 43 14 heat kernel is the transition density of the corresponded Markov 19 21 In the previous section, the classical heat kernel depend 37 Informally, heat operator satisfies parabolic Harnack inequality itssolutions, solutions, further 37 Informally, aadensity heat operator satisfies parabolic Harnack inequality ififits further ry, see (Kassman, 2007). 16 subject will not be discussed here. 13 In general, any heat operator is a Markov operator of some Markov 13 In general, any heat operator is a Markov operator of some Marko rmally, a heat operator satisfies parabolic Harnack inequality if its solutions, further 25 is a (weak) solution of the h a random variable. Actually, is the transition of the Brownian motion on . uous. For further information about Harnack inequality including its variance and 22 intrinsic distance i.e. the Euclidean distance on , and another is the dimension 18 operators behaves similarly to the Gaussian kernel. 39athat inequality issatisfies thati.e. such solutions always admit continuous representative soincluding one may assum Actually, pbehaves isinequality the transition density of the Brownian Informally, heat operator parabolic Harnack ifpossible its solutions, further 40 40 they they are areboundary, continuous. continuous. For For further further information information about about Harnack Harnack inequality inequality including its its va esentative so one may assume 18 operators to the Gaussian kernel. 24 kernels. Consider aquestion heat operator where is aMarkov fixed constant. It to isv(C ero and . further Then is aconstant. probability density of 15 2012). Therefore, itsimilarly isofof touniform study processes via 20 Gaussian Behavior 22 intrinsic distance i.e. the Euclidean distance on , Processes and anoth 38 away from the boundary, are roughly constant. One consequence of uniform parabolic Harnack 38 away from the are roughly One consequence of parabolic Harnack ality if 37 itsvariance solutions, computed up to sets of measure zero. 17 One possible is that whether or not heat kernels associated 14 heat kernel is the transition density the corresponded Markov Processes 14 heat kernel is the transition density the corresponded Markov the boundary, are roughly constant. One consequence of uniform parabolic Harnack In general, any heat operator is a Markov operator of some Markov processes and the ssman, 23 21 underlying space It isconsequence interesting toabout know whether this behavior happens forbe( 19382007). 40 they are continuous. For further information Harnack inequality including its varian away from the boundary, are roughly constant. One uniform parabolic Harnack 41 41 history, history, see see (Kassman, (Kassman, 2007). 2007). equality including its 39 variance and 19solution motion on .of 16 subject will not discussed here. om variable. Actually, is the transition density of the Brownian motion on .be In the previous section, the classical heat kernel depends on two objects, 23 underlying space It is interesting to know whether this 39 inequality is that such solutions always admit continuous representative so one may assume that inequality is that such solutions always admit continuous representative so one may assume that uniform parabolic Harnack 25 is a (weak) of the heat equation associated if and only if so 18 operators behaves similarly to Gaussian kernel. 15 2012). Therefore, it is possible to study Markov processes via heat equatio 15 2012). Therefore, it is possible to study Markov processes via heat equat 26 Therefore, the heat kernel assoc s that such solutions always admit continuous representative so one may assume that atisfies parabolic Harnack inequality ifkernels. its further any Informally, a42 heat operator satisfies parabolic kernel is the transition density of thesolutions, corresponded Processes (Chenand &question Fukushima, Consider aMarkov heat operator where isand a fixed constant. Itdimension isaheat easy 41 history, see (Kassman, 2007). 2039 Gaussian Behavior inequality isthat that such solutions always admit continuous representative so one may assume that 42 20 Gaussian Behavior Inheat general, heat operator is24 aare Markov operator of some Markov processes the 17 One possible is that whether or not k 22 intrinsic distance i.e. the Euclidean distance on , another is the 24 kernels. Consider a heat operator where is fixed 40 they continuous. For further information about Harnack inequality including its variance and 40 they are continuous. For further information about Harnack inequality including its variance and ve so one may assume In general, any heat operator is a Markov operator 19 16 subject will not be here. 16 classical subject will not bediscussed discussed here. For further information about Harnack inequality including its variance and yntinuous. constant. One consequence of uniform parabolic Harnack 2012). Therefore, it is possible to study Markov processes via heat equations. However, that 21 In the previous section, the heat kernel depends on two objects, one is the Harnack inequality if its solutions, further away from 42 40 they are continuous. For further information about Harnack inequality including its variance and 43 43 21 In the previous section, the kernel dep 25 23 is a17 (weak) solution heat equation associated to ifclassical and only if solves is the transition density of the corresponded Markov Processes (Chen & Fukushima, 18 operators behaves similarly to the Gaussian kernel. 27 As aor generalization of thisassoc underlying space Itthe is(weak) interesting to whether this behavior happens for 41 history, see (Kassman, 2007). 41 see (Kassman, 2007). 26 Therefore, the heatof kernel associated toisknow isthe ,to which is yrnel including its variance and history, 20 Gaussian Behavior One possible question is that whether not heat kernels 17 One question that whether or not heat kernels assoc 25 ispossible aand solution of heat equation associated ifpro an of some Markov processes and the heat kernel is heat the (Kassman, 2007). ays admit continuous representative so one may assume that subject not be discussed here. 43 2241 will intrinsic distance i.e. the Euclidean distance on , another is the dimension of the history, see (Kassman, 2007). the are roughly constant. One consequence 22 intrinsic distance i.e. the Euclidean distance on , and an Therefore, it isboundary, possible to study Markov processes via heat equations. However, that 19 28 upper and lower bound. Nash 24 kernels. Consider a heat operator where is a fixed constant. It is easy 42 42 21 In the previous section, the classical heat kernel depends on two objects, 18 operators behaves similarly to the Gaussian kernel. 18 variance operators behaves similarly toofthe Gaussian kernel. rmation about Harnackspace inequality including its and One possible question is that whether or not heat kernels associated to other heat transition density the corresponded Markov Processes 23 underlying It is interesting to know whether this behavior happens for other heat 27 As a generalization of this property, one might instead ask whether a heat kern 42 26 Therefore, the heat kernel associated to is , which is again 23i.e. underlying space Iton know whether this will not be discussed here. 20 29is interesting (Aronson, 1967, 1968, 1971) co 43 43 of uniform parabolic Harnack isdistance that such 26 Therefore, the heat kernelassociated associated to Gaussian is to 22 inequality intrinsic the Euclidean distance , to and another isBehavior the dimensio 19 25 is a (weak) solution of isthe heat equation and only if1964, solves 19 operators behaves similarly to the Gaussian kernel. 28 orupper andwhere lower bound. Nash (Nash, 1958), Moser (Moser, 1961, 1967 24 Consider a heat operator constant. It30 isTherefore, easy toand see that 43 kernels. (ChenaIn &fixed Fukushima, 2012). itclassical isifYao(Li possible toYao, 24 kernels. Consider a heat operator where is a fix One possible question is that whether not heat kernels associated to other heat 21 the previous section, the heat kernel depe Li & 1986) Gaussian Behavior 23 Asunderlying space Itproperty, is interesting to know whether this behavior fore solutions always admit27continuous representative so1968, 20 Gaussian Behavior a 20 generalization of this one of might instead ask whether ainstead heathappens kernel has 29 (Aronson, 1967, 1971) concluded that this is true for uniformly elliptic ope 27 As a generalization this property, one might ask wheth ors behaves similarly to thesolution Gaussian kernel. 22 intrinsic distance i.e. the Euclidean on , Iton and ano 31 Ricci curvature. However, Liif 25 is a (weak) of the Therefore, heat equation associated to if and only if solves . associated study processes via heat(Moser, equations. However, is a Markov (weak) of the heat equation toagain 26 the heat associated tosolution isthe , which isis 21 In25 the previous section, classical heat kernel depends two 21 In& the previous section, the classical heat kernel depends on two 24 kernels. Consider akernel heat operator where isLaplacians adistance fixed easy upper and lower bound. Nash (Nash, 1958), Moser 1961,constant. 1964, 1967), and one may assume that 28 they are continuous. For28 further Gaussian Behavior 30 Li and Yao(Li Yao, 1986) extended this result to onwhether manifolds w upper and lower bound. Nash (Nash, 1958), Moser (Moser, 19 32 should be replaced with the vol 23 underlying space It is interesting to know this that subject will not be discussed here. 22 intrinsic distance i.e. the Euclidean distance on , and another is the d 22a heat intrinsic distance i.e. the Euclidean distance on , and another is the 29the (Aronson, 1967, 1968, 1971) concluded that this is true for uniformly elliptic operator. In the previous section, classical kernel depends on two objects, one is the 25 is (weak) solution of the heat equation associated to if and only if solves 31 As Ricci curvature. Lithe and Yao also showed that the term isisuniform noakerne lon 27associated a generalization ofHowever, this property, one might instead ask whether heat kernel ha 292426 (Aronson, 1967, 1968, 1971) concluded that this isa true for 26 Therefore, the heat kernel to is , which is again Gaussian. Gaussian Behavior 33 Gaussian behavior of heat kernels. Consider a heat operator where fixe Therefore, heat kernel associated to is 23 space ItItisis interesting toto know whether this 23Yao(Li underlying space interesting know whether thisbehavior behavior hap 30 28 32 Li and & 1986) extended this resultMoser Laplacians on manifolds withhap noo distance i.e. thetheEuclidean distance onunderlying , Yao, and another is volume the of the should be replaced with the measure. The following definition gives the upper and lower bound. Nash (Nash, 1958), (Moser, 1964, 1967), an 30 Li and Yao(Li &dimension Yao, extended this1961, result to Laplacians Inintrinsic the previous section, classical heat kernel depends on two objects, one is1986) the 34 Santi Tasena / Burapha Sci. J. 19 (2014) 1 : 169-177 25 is a (weak) solution of the heat equation associated to if 172 24 kernels. Consider a heat operator where is a fixed constant. 24 kernels. Consider a heat operator where is a fixed constant. Ricci curvature. However, Li and Yaoto thatfor the term is no 26 33 Therefore, the heat kernel associated is , instead which is again 27 As athe generalization of31this property, might ask whether aalso heat kernel has Gaussian space It is interesting to Gaussian know this behavior happens for other heat behavior heat kernel. 27instead As adimension generalization ofshowed this one might ask wc 29 (Aronson, 1967, 1968, 1971) concluded that this is property, true uniformly elliptic operator 31 Ricci curvature. However, Li and Yao also showed thatlonger the ter c underlying distance i.e. Euclidean distance on , one andwhether another isof the of the 32 should be replaced with the volume measure. The following definition gives the curre 28 Consider upper Itand bound. Nash (Nash, 1958), Moser (Moser, 1961, 1964, 1967), and Aronson kernels. heat operator where is ashould fixed constant. Itother is easy tovolume see 25 isisa32 (weak) solution of the heat equation associated toto manifolds iffollowing ififn 25 Yao(Li abehavior (weak) solution of the heat equation associated ifand andonly only 34 Liwhether 28 upper and lower bound. Nash (Nash, 1958), (Moser, & Yao, 1986) extended this result to that Laplacians with be replaced with the measure. defin ying space isa lower interesting to30 know this happens for heat 27 Asand a generalization of this property, one might instead ask aMoser heat kernel ha 26 Therefore, the heat kernel associated towhether isonThe 33 Gaussian behavior of heat kernel. 29 (Aronson, 1967, 1968, 1971) concluded that this is true for uniformly elliptic operator. Later on, 29 1967, 1971) that this is true for unic is a (weak) solution of the heat31equation if (Aronson, and only if solves . the term Ricciassociated curvature. However, Libehavior and Yao also that no longer 33 to Gaussian of1968, heat showed kernel.concluded . c distance .is nothing but the Euclidean distance on nce on 1 Definition 5. A heat kernel is said to have Gaussian upper b , the Sobolev space of order one. The t kernel associated to is defined by , the Sobolev space of order one. The2 ed to is defined by such that all x,y in underlying space andupper t>0, 1 1 Definition Definition 5.5. AAheat heatfor kernel kernel isisthe said said totohave haveGaussian Gaussian upperbou bo 9 9 which which will will be be called, called, for for the the rest rest of of this this work, work, the the classical classical heat heat kernel. kernel. Sobolev norm norm , the Sobolev space of order one. The 2 2density such that that forallall x,y x,ydistribution indistribution inthe the underlying underlying spaceexist and andt>0, t>0, 10 10 Denote Denote the theGaussian Gaussiankernel kernel i.e.the the of ofthe thesuch Gaussian(Normal) Gaussian(Normal) with withifspace 3 i.e. Adensity heat kernel is said tofor have Gaussian lower bound there constants olev norm 6 that for all x, y ini.e. the and t>0,density 11 11 for mean mean zero and variance variance . .classical Then Then 4heatsuch i.e. underlying isisaspace aprobability probability densityof of alled, thezero rest and of this work, the kernel. heat kernel. One possible question is that whether or not 12 12 theaarandom randomvariable. variable. Actually, Actually, isisof the the transition transition density densityof ofthe theBrownian Brownian motion motionon on 1 . . Definition 5. A heat kernel Gaussian kernel i.e. the density the Gaussian(Normal) distribution with he but Gaussian(Normal) distribution with heat kernels associated to other heat operators behaves . ng the Euclidean distance on 2 13 13 In In general, general, any any heat heat operator operator is is a a Markov Markov operator operator of of some some Markov Markov processes processes and and the the such that for all x,y in . einition Euclidean distance on heat kernel density is saidofto havei.e. Gaussian isupper bound ifdensity there exist variance . isThen a probability of constants i.e. 5. A a probability A heat kernel is said to have Gaussian lower bound if th 3 associated to is defined by 14 14 heat heat kernel kernel is the transition density density ofspace the the corresponded corresponded Markov Processes(Chen (Chen&&Fukushima, Fukushima, associated toall istransition defined by similarly tothe the Gaussian kernel. such that for x,y in the. density underlying and t>0, ut the Euclidean distance on ble. Actually, is isthe transition of of the Brownian motionMarkov on . Processes he Brownian motion on . such A that for all x, yequations. inisthe underlying space t>0, 3 3 4 processes Aheat heat kernel issaid said toHowever, to have haveGaussian Gaussian lowerbound boundififthere the 15 15 2012). 2012). Therefore, Therefore, is possible possible toto study studysome Markov Markov processes via viakernel heat heat However, that thatandlower heat operator istoaititMarkov operator Markov processes andequations. the associated isisand defined by orral, ofany some Markov processes the 5 ofHere volume measure and isthe the intrinsic distance. 4 4is the such such that thatfor forallallx,x, y yininthe underlying underlying space spaceand andt>0, t>0, 16 16 subject subject will will not not be be discussed discussed here. here. he transition density ofBehavior the corresponded Markov Processes (Chen Fukushima, 6 A heat kernel is&said satisfy Gaussian estimates has intrinsic both Gaussian upper Here ν istothe volume measure and heat ρif isit the arkov Processes (Chen & Fukushima, Gaussian 17 One One possible possible question question isis that that whether whether or or not not heat heat kernels kernels associated associated to to aother other heat f17 this work, the classical heat kernel.processes ork, the classical heat kernel. 7 and Gaussian lower bound. A heat operator or Dirichlet form is said to satisfy Ga ore, it is possible to study Markov via heat equations. However, that es via operators heat equations. However, that distance. In the previous section, the classical heat kernel 18 18 operators behaves behaves similarly similarly to to the the Gaussian Gaussian kernel. kernel. kernel i.e. the density of the Gaussian(Normal) distribution with 3 A heat kernel is said t 8 estimates if its associated heat kernel exists and satisfies Gaussian estimates. .be the density ofhere. the Gaussian(Normal) distribution with discussed is work, the classical heat kernel. 19 19 A heat kernel is said to satisfy Gaussian estimates depends on two objects, one is the intrinsic distance 9 4 such that for all x, y in the under i.e. is density i.e.that a probability density of ssible question isof orGaussian nota probability heat kernels associated other heat Akernels heat kernel is said toishave lower bound ifoftheretoexist constants eat associated towhether other heat el the density the Gaussian(Normal) distribution with 20 20i.e. Gaussian Gaussian Behavior Behavior 5 Here is the volume measure and is the theLebesgue intrinsiclower distance.as their v 10 For heat operators in Euclidean space with measure if it has both Gaussian upper bound and Gaussian the transition density of the Brownian motion on . the Euclidean distance and density another isofthe ition density ofythe the Brownian motion on similarly to kernel. hves that for all i.e. x, in Gaussian the underlying and., t>0, i.e. isspace aon probability 21 21 In In the the previous previous section, section, the the classical classical heat heat kernel kernel depends depends on on two two objects, objects, one one is is the the 6 A heat kernel is said to satisfy Gaussian estimates if itredu ha 5 5 Here Here is is the the volume volume measure measure and and is is the the intrinsic intrinsic distance. distance. ator is a Markov operator of some Markov processes and 11 measure , the bound. A heat foroperator all for someform fixedis said constant Markov operator some Markov processes and or a Dirichlet to satisfy . This dimension n of the underlying space interesting transition density ofofthe Brownian motion ondistance . It isthe 22 22 intrinsic intrinsic distance distance i.e. i.e. the the Euclidean Euclidean distance on on , , and and another another is is the the dimension dimension of of the the Gaussian lower A heat(Nash, operator or a Dirichlet form 6Fukushima, 6 7 typeand AAheat heatfound kernel kernel isbound. said said toto satisfy satisfy Gaussian Gaussian estimates estimates ififitithas hasb tycorresponded of the corresponded Markov Processes (Chen Markov Processes (Chen &happens Fukushima, Gaussian Behavior 12 the& same of estimates inisthe work of Nash 1958), Moser 1961, Gaussian estimates if heat its associated heat kernel exists(Moser, is23a Markov of some Markov processes and the tooperator know whether thisItItheat behavior for other heat 23 underlying underlying space space is is interesting interesting to to know know whether whether this this behavior behavior happens happens for for other other heat heat 8 estimates if its associated kernel exists and satisfies Gaussian ei 7 7 and and Gaussian Gaussian lower lower bound. bound. A A heat heat operator operator or or a a Dirichlet Dirichlet form form to study Markov processes via equations. However, that 13 depends and (Aronson, processes viaclassical heatone equations. However, that previous section, heat on Aronson two objects, one 1967, is the1968, 1971). As a generalization of Euclidean dimen depends on two the objects, is thekernel fMarkov the corresponded Markov Processes (Chen &1967), Fukushima, and satisfies Gaussian estimates. kernels. Consider a heat operator where 24 24 kernels. kernels. Consider Consider a a heat heat operator operator where where is is a a fixed fixed constant. constant. It It is is easy easy to to see see that that 9 8 is estimates estimates ififitsitstoassociated associated heat heat kernel kernel exists existsand and satisfies satisfies Gaussian Gaussian estim est 14 another one 8might be dimension tempted define (local) the underlying space ceanother i.e. theisEuclidean distance on the of thethe 5 dimension Here isofthe volume measure an dtudy the dimension the , and Markov processes via heatof equations. However, that For heat operators in Euclidean space with 10 For heat operators in Euclidean space with the Lebesg 9 9 that whether or not heat kernels associated to other heat u is a (weak) is a fixed constant. It is easy to see that 25 25 is is a a (weak) (weak) solution solution of of the the heat heat equation equation associated associated to to if if and and only only if if solves solves . . hether or not heat kernels associated to other heat 6 A heat kernel is said to 15 or any other terms of similar nature. However, this limit usually does no ce It is interesting to know whether this behavior happens for other heat happens forand other isheat ethisisbehavior the volume measure the intrinsic distance. 10 10 For For heat heat operators operators in in Euclidean Euclidean space space with with the the Lebesgue Lebesgu Gaussian kernel. the Lebesgue measure as their volume measure n kernel. 7 and Gaussian lower bound. A solution ofsaid thekernels associated to11Lheat ifif measure and der a heat operator where is fixed constant. is both easy to see that for all for some fixed con 16 a to regardless of whether the ,heat kernel has Gaussian estimates. Even when the limit is repla a fixed constant. is easy to seeequation that A heat kernel toheat satisfy Gaussian estimates it It has Gaussian upper bound at whether or notItisheat associated other 26 26 Therefore, Therefore, the the heat heat kernel kernel associated associated to to is is , , which which is is again again Gaussian. Gaussian. 8for estimates iffor its heat 17 oriflimsup or liminf, itsame still the asfor to zero isassociated not the same as ke th forconverges all forsome some 11 11 measure measure all all for some fixed fixed consta const Gaussian lower A heat operator athe Dirichlet form is, ,possible said of tothat satisfy Gaussian ussian kernel. 12 the type estimates found in the work of Nash (Nash, 1958 solution ofonly the heat equation associated to and only if solves . limit if Gaussian and . Therefore, usolves solves heat kernel onlyifBehavior ifbound. 18 as converges to infinity. In other words, there is no universal definition of dimens ussian Behavior 9 mates ifAs itsaaassociated heat kernel and one satisfies Gaussian estimates. fixed constant cheat > 0kernel .found This reduced to the of same typeAs ofa generaliza 1967), and Aronson (Aronson, 1967, 1968, 1971). 121213 the the same same type type ofofestimates estimates found in inthe the work work ofNash Nash (Nash, (Nash, 1958), 1958),M 27 27 classical As generalization generalization of ofthis thisexists property, property, one might might instead ask ask whether whether aaheat kernel has has Gaussian Gaussian the heatdepends kernel depends on two19objects, oneinstead is the general. Nevertheless, Gaussian estimates still imply a property related to the ical heat kernel on two objects, one is the 10 For heat operators inofgro Eut associated to L is Gaussian. which heat kernel associated to is ,,1967), which ismight again Gaussian. Gaussian Behavior 14Moser one be tempted to define the (local) dimension 13 13 1967), and and Aronson Aronson (Aronson, (Aronson, 1967, 1967, 1968, 1968, 1971). 1971). As As a a generalizatio generalizati , which is again 28 28 upper upper and and lower lower bound. bound. Nash Nash (Nash, (Nash, 1958), 1958), Moser (Moser, (Moser, 1961, 1961, 1964, 1964, 1967), 1967), and and Aronson Aronson estimates found in the work of Nash (Nash, 1958), Moser 20 the volume measure called doubling property(Sturm, 1996). ean distance on , and another is the dimension of the ance on , and another is the dimension of the classical heatoperators kernel on two objects, one isthe the heat in Euclidean space with Lebesgue measure asany their volume 11 measure , nature. 1414 might might bebe1961, tempted tempted to to define define the the (local) (local) dimension dimension ofof the th 29 29For(Aronson, (Aronson, 1967, 1967,depends 1968, 1968, 1971) 1971) concluded concluded that that this this isone isone true true for for(Moser, uniformly uniformly elliptic elliptic operator. operator. Later Later on, on, 15 or other terms of similar th 21 other 1964, 1967), and Aronson (Aronson, 1967,However, to know whether this behavior happens for other heat of this property, one might instead ask whether a heat kernel has Gaussian is again Gaussian. As a generalization of this property, oation know whether this behavior happens for heat kesting whether a heat kernel has Gaussian distance on , and another is the dimension of the 30 30 Li Li and and Yao(Li Yao(Li & & Yao, Yao, 1986) 1986) extended extended this this result result to to Laplacians Laplacians on on manifolds manifolds with with nonnegative nonnegative 22It for Doubling Property 12 same type ofestimates. estimates found sure , where for allIt is easy some constant .Asany This reduced to 15 1516 oror any other other terms terms of similar similar nature. nature. However, However,Even this this of whether the heat kernel has Gaussian isinstead a fixed constant. isheat easy tofixed see regardless that er bound. Nash (Nash, 1958), Moser (Moser, 1961, 1964, 1967), Aronson 1968,and 1971). a generalization ofofthe Euclidean dimensions, where is amight fixed constant. to see that oser, 1964, 1967), and Aronson one ask whether afor kernel ng to1961, know whether this behavior happens other heathasthat 23Yao Actually, doubling property isisano property ofcorrect the1967), underlying spaces and(Aronson, not of the Di 31 31 Ricci Ricci curvature. curvature. However, However, Li Li and and Yao also also showed showed that the the term term no longer longer correct and and 13 and Aronson 17elliptic limsup or liminf, ittempted still possible that the limit asestimates. converges to zer 7, 1968, 1971) concluded that this isonly true forofuniformly operator. Later on, same type of estimates found inif the work Nash (Nash, 1958), Moser (Moser, 1961, 1964, 16 regardless regardless ofof whether whether the theby heat heat kernel has has Gaussian Gaussian estimates. Even Even w1w uniformly elliptic operator. on, at equation associated to ifLater and solves one might tokernel define the (local) dimension of on associated to if upper only solves . has where is aand fixed constant. Itifis easy to16 see that 24measure. forms. ItThe been. studied as a be subject itself. The book of Heinomen (Heinomen, 200 Gaussian and lower bound. Nash (Nash, 1958), 32 32 should should be be replaced replaced with with the the volume volume measure. The following following definition definition gives gives the the current current form form of of 14 one might be tempted to de & Yao, 1986) extended result1968, to Laplacians on manifolds with nonnegative 18 as converges to infinity. In other words, is no to universa 7), andon Aronson (Aronson, 1967, 1971). As astarting generalization of Euclidean dimensions, 17 17 limsup limsup orThere liminf, liminf, still still possible possible that that the thelimit limit asthere converges converges to zero zero acians manifolds withthis nonnegative 25 good point.or areitit two closely related versions of as doubling property, one is i 33 33 Gaussian Gaussian behavior behavior of of heat heat kernel. kernel. quation associated to if and only if solves . Moser (Moser, 1961, 1964, 1967), and Aronson (Aronson, general. Gaussian estimates adoubli prop theand underlying space asas measure orimply e.ted However, Li no andlonger Yao showed that term no longer correct and might to also define theis, which (local) dimension ofconverges the underlying space 15 any ted 181819 asisas converges toNevertheless, toinfinity. infinity. other other words, words, there there isspace isor no no universal universal the termbe tempted is correct and is space again 26 the toGaussian. be doubling, another is for theInIn volume of thatstill to beother s34 , which again Gaussian. 34to is 1967, 1968, 1971) concluded that this is true for uniformly 20 that the volume measure called doubling property(Sturm, 1996). ceddefinition with the gives volume The following definition gives the current form 1919 general. general. Nevertheless, Nevertheless, Gaussian Gaussian estimates estimates still imply imply a apropert prope 27 nature. turns out thethis latter implies the of former while the former implies the existence of ke a ng themeasure. current form of 16exist regardless ofstill whether the heat or any other terms similar However, limit usually does not to is instead ,of which is again Gaussian. any other terms of similar nature. However, this limit rty, one might instead ask whether a heat kernel has Gaussian might ask whether a heat kernel has Gaussian 21 vior of heat kernel. 28 measure which is doubling(Luukkainen & Saksman, 1998). Therefore, only doubling pr 20 20 the the volume volume measure measure called called doubling doubling property(Sturm, property(Sturm, 1996). 1996). ellipticthe operator. Later on, LiGaussian and Yao(Li & Yao, 1986) 17 bylimsup or liminf, it still possible rdlessMoser of whether heat kernel has estimates. Even when the limit ishere. replaced Nash, 1958), Moser (Moser, 1961, 1964, 1967), and Aronson 29 of volume measures usually will be discussed does not exist regardless whether the heat 958), (Moser, 1961, 1964, 1967), and Aronson 22 Property 21 21 one might instead ask whether a heat kernel has Gaussian 18limitasofDoubling converges to infinity. In extended this result tothe Laplacians on converges manifolds with up or liminf, it still possible that limit as to zero is not the same as the 30 cluded that this is true for uniformly elliptic operator. Later on, at this is true for uniformly elliptic operator. Later on, 23 Actually, doublingestimates. property is a property of theisunderlying 22 22 Doubling Doubling Property Property kernel has Gaussian Even when the limit h, 1958), Moser (Moser, 1961, 1964, 1967), and Aronson 19 general. Nevertheless, Gaes 31there Definition 6.also A non-zero Borel measure on a metric is doubling if there converges to infinity. In other words, noYao universal definition of dimension in space nonnegative Ricci curvature. However, Li is and ended this result to Laplacians on manifolds with nonnegative is result to Laplacians on manifolds with nonnegative 24 forms. It has been studied as a subject by itself. The book of Hein 23 23 on, Actually, Actually, doubling property property isitisthe astill aproperty property of the thethe underlying underlying spa sp ed that this Nevertheless, is true for uniformly elliptic operator. Later 20 of volume called bydoubling limsup or liminf, possibleofmeasure that 32 longer constant suchreplaced thatrelated for any and . doub eral. Gaussian estimates still imply aand property to the growth that the is no correct nd Yao also showed showed that the term term is no longer correct and 25 good starting point. There are two closely related versions of dou lso showed that the term is no longer correct and 24 24 forms. forms. It It has has been been studied studied as as a a subject subject by by itself. itself. The The book book of of Heinom Heino 33 ed this result to Laplacians on manifolds with nonnegative 21is not the same as the limit volume doubling property(Sturm, 1996). limit t convergesand to zero me measure. Thecalled following definition gives current form ofstarting 26 space to Saloff-Coste(Grigor’yan beaspoint. doubling, another is for related the volume measure of t ure. Themeasure following definition should bethe replaced with the volume measure. The 34 theform Grigor’yan and & Saloff-Coste, 2005) extended the conc 2525of good good starting point. There Thereare are two two closely closely related versions versions ofof doubli doub Yao also showed that termgives isthe nocurrent longer correct and 22 t converges to infinity. In other words, there is no as 35 current doubling property that of remote and -anchored balls. They showedthe thatformer doubling pr turnstoto that the latter implies the former while 2626 space space toout be bedoubling, doubling, and andanother another is for for the the volume volume measure measure of ofimp tha th definition givesgives the current form of27Gaussian measure. Thefollowing following definition the form of 23 is Actually, doubling prope Doubling 36 Property of 27 remote and S-anchored balls together withthe another property called volumeimplie comp measure which is doubling(Luukkainen & Saksman, 1998). There universal definition of dimension in general. Nevertheless, 2728 turns turns out out that that the the latter latter implies implies the former former while while the the former former impli 24 forms. It has been studied as a s of heat kernel.is a property Actually,behavior doubling property the underlying spacesmeasures and not for of the Dirichlet 37 ofcondition imply doubling property allbe balls providedhere. that the set S1998). is fully acce of volume will discussed 282829 measure measure which which is is doubling(Luukkainen doubling(Luukkainen && Saksman, Saksman, 1998). Therefo Theref Gaussian(Heinomen, estimates imply related to the 25 isaaproperty good starting point. There are t 6 still ms. It has been studied5.asA aheat subject The book of Heinomen 2001) Definition kernelbyp itself. is said to have Gaussian 30 2929 ofofvolume volume measures measures will willbebediscussed discussed here. here. 26 the space to on be adoubling, and anoth growth ofAthenon-zero volume called doubling property d starting point. twoexist closely related of doubling property, one measure is for 31 Definition 6. Borel measure metric space 30 30 c1, cversions > 0 such that upper There bound are if there constants 2 27 Itturns out that the latter implies ce to 5.be Adoubling, and another forGaussian the volume measure that tosuch be that doubling. (Sturm, 1996). Definition Definition 6.space 6.AA non-zero non-zero Borel Borel measure and onona ametric metricspace space i constant for measure any tion heat kernel said to is have upper bound ifof there exist constants for all x, y inisthe underlying space and t >31031,32 28 measure which is doubling(Luu s out that the latter implies the former while the former implies the existence of a Borel 33 such that for all x,y in the underlying space and t>0, 3232 constant constant such suchthat thatfor forany any and and 29 of volume measures will be disc Grigor’yan and Saloff-Coste(Grigor’yan & Saloff-Coste, 2 sure which is doubling(Luukkainen & Saksman,33 1998). Therefore, only doubling property 3334 30 Doubling Property doubling property to Saloff-Coste(Grigor’yan that of remote and -anchored balls. They sh olume measures will be discussed here. 343435 Grigor’yan Grigor’yan and and Saloff-Coste(Grigor’yan Saloff-Coste, Saloff-Coste, 2005 200 31 Definition & 6.&A non-zero Borel Actually, doubling property is a property of the 36 of remote and S-anchored balls together with another property 3535 doubling doublingproperty propertytotothat thatofofremote remoteand and -anchored -anchoredballs. balls. They Theyshow sho 32 aconstant such that for an inition 6. A non-zero Borel measure on a metric space isand doubling if thereballs exists imply doubling for all balls provided that the 363637 ofofcondition remote remote and S-anchored S-anchored balls together together with with another another property property c underlying spaces andproperty not of the Dirichlet forms. It 33 stant thattoforhave anyGaussianand . property A heat kernel such is said lower bound 37 if37therecondition exist constants condition imply imply doubling doubling property for for all all balls balls provided provided that that the the set se 34 has been studied as a subject by itself.Grigor’yan The book ofand Saloff-C is said the underlying A heat kernel hat for all x, y in spacep and t>0,to have Gaussian lower 35 of property Grigor’yan andif Saloff-Coste(Grigor’yan extended the concept Heinomen (Heinomen, 2001) is doubling a good starting point.to that of rem bound there exist constants c3, c4 & > 0Saloff-Coste, such that for 2005) 36 of remote and S-anchored bal bling property to that of remote and -anchored balls. They showed doubling There that are two closelyproperty related versions of doubling all x, y in the underlying space and t > 0, 37 condition imply doubling prope emote and S-anchored balls together with another property called volume comparison property, one is for the space to be doubling, and another dition imply doubling property for all balls provided that the set S is fully accessible. is the volume measure and is the intrinsic distance. A heat kernel is said to satisfy Gaussianสันestimates has both Gaussian upper bound1 : 169-177 ติ ทาเสนาif/itวารสารวิ ทยาศาสตร์ บูรพา. 19 (2557) aussian lower bound. A heat operator or a Dirichlet form is said to satisfy Gaussian tes if its associated heat kernel exists and satisfies Gaussian estimates. 173 Gaussian estimates. Evenofwhen the limit is replaced by 71). As aweighted generalization Euclidean dimensions, 2asasasconverges measures with finite unbounded weight Moshimi and Tesei(Moshimi s,1810 there is no converges universal toclosed definition infinity. of In dimension other inand there isfunctions. no definition of dimension 10 , S is closed and null, and Assume the following conditions hold. 10 , , S S is is closed and and null, null, and and ..universal . universal Assume Assume the the following following conditions conditions hold. 2 limit weighted measures with finite weight Moshim 18 as converges to to infinity. infinity. In In other other words, there there is is no no universal definition definition ofunbounded of dimension dimension inhold. inin&functions. 10 ,words, Sas is closed and null, and . of Assume the following condi imit converges zero is not the same aswords, the Using this fact when is singleton, Grigor’yan Saloff-Coste proved doubling property al) 3dimension of the underlying space Tesei, 2007) also used this fact to show doubling property of weighted Lebesgue measures with mates 19 still general. imply a property Nevertheless, related to Gaussian the growth estimates of still imply a property related to the growth of 3 Tesei, 2007) also used this fact to show doubling property of weighted 19 general. general. Nevertheless, Nevertheless, Gaussian Gaussian estimates estimates still still imply imply a property a property related related to the the growth growth of of 11 a) The set S satisfies -skew condition i.e. for any and there exists such that 11 11 a) a) The The set set S S satisfies satisfies -skew -skew condition condition i.e. i.e. for for any any and and there there exists exists such such that that rds, there ismeasures no universal ofa)dimension infunctions. weighted with definition finite11 unbounded Moshimi and & The weight set S satisfies -skew condition i.e.Tesei(Moshimi for any and there exists nature. However, this limit usually doesproperty(Sturm, not exist 4 the unique-singularity weight To4 be property precise, they provedweight doubling property measures y(Sturm, 20Tesei, the 1996). volume measure called doubling property(Sturm, 1996). unique-singularity functions. Toforbe precise, they proved doubli 20 the volume volume measure measure called called doubling doubling property(Sturm, 1996). 1996). imates still imply a used property related to the growth of 2007) also this fact tofunctions. show doubling of weighted Lebesgue measures with 5estimates. where is aremotely positive number less than the dimension ofathe underlying Euclidean 21 sian when the limit replaced by 21 5 there where is positive number less thanspace. the dimension the for und b) The function is constant i.e. there exists constant 12 such that for any 12 12 b) b)The TheEven function function is isfunctions. remotely remotely constant i.e. i.e. there exists exists aaaconstant constant such that that for for any rty(Sturm, 1996). unique-singularity weight ToThe be precise, they doubling property for measures b) function is proved remotely constant i.e.such there exists aany constant 12 is constant suchofthat ling 22 Property Doubling Property 6 Later, the fully accessible assumption was weaken to that of -skew condition and the uniqueas converges to zero is not the same as the limit 22 13 Doubling Doubling Property Then is for theisvolume measure space to6 the beProperty doubling. isunderlying doubling if Euclidean and only if was there exists a constant Later, the fully assumption weaken to that of -skew c 13 13 where a positive number than dimension of theµaccessible space. 13of thatless 3 3 erty 23 of the underlying Actually, spaces doubling and property not of the is Dirichlet a property of the underlying spaces and not of the Dirichlet 7 singularity assumption was removed yielding the following result. there is no universal definition of dimension in 23 Actually, Actually, doubling doubling property property is a is property a property of of the underlying underlying spaces spaces and and not not of of the the Dirichlet Dirichlet 7 singularity assumption was removed yielding the following ubling Property Itfully turns out that the latter implies the weaken former the such that for and 14 Then doubling and only there exists aaconstant constant such that for any and 14 14 Then Then isisisdoubling doubling ifififassumption and and14 only only ififThen ifthere there exists exists awhile constant such such that that for forany any any and and such that forresult. Later, the accessible was to that of only -skew condition and the uniqueis doubling if and if there exists a constant any 8 8is self. 24still The forms. book of has Heinomen been studied (Heinomen, asubject subject 2001) byof itself. a The The book of Heinomen (Heinomen, 2001) 3Heinomen es imply aIthas property related to growth 24 forms. forms. It has Itunderlying been been studied studied asand as aassubject athe by by itself. itself. The book book of of Heinomen (Heinomen, (Heinomen, 2001) 2001) is is a isa a operty of the spaces not of the Dirichlet singularity assumption was removed yielding the following result. former implies the existence of a Borel measure which is 9 Theorem 1.(Tasena, 2011) Let be a doubling measure on a metric space , 9 Theorem 1.(Tasena, 2011) Let be a doubling related 25 good versions starting of doubling point. There property, are two one closely is for the related versions of doubling property, one is for themeasure on a metric s turm, 1996). 25 good good starting starting There There are are two two closely closely related related versions of of doubling doubling property, property, oneone is is forfor thethe 15 15 15 itself. The bookpoint. ofpoint. Heinomen (Heinomen, 2001) is a versions 15 k -skew subsets of Euclidean doubling (Luukkainen & Saksman, 1998). Therefore, only Examples of 10 , S is closed and null, and . Assume the following conditions hold. volume space measure tobe be doubling, ofdoubling that space and to another isfor for the It volume measure ofthat that space to bedoubling. doubling. 10 Sspaces isinclude closed and null, and . Assume the fo Theorem 1.(Tasena, 2011) Let besubsets adoubling. doubling measure on a metric space ,subspaces, 26 space space to to be doubling, doubling, and another another is is for the the volume volume measure of of that space space to to be be doubling. It include ItIt spaces 16 Examples of -skew subsets of Euclidean spaces include proper subspaces, spheres, 16 Examples Examples of ofand -skew -skew subsets of Euclidean Euclidean spaces include proper proper subspaces, spheres, spheres, ye2616 related versions of property, one isof for the 16 Examples of,measure -skew subsets of Euclidean spaces proper subspace 11 a) The set S satisfies -skew condition i.e. for any and there exists such that 27 while turns the former out that implies the latter the existence implies the of former a Borel while the former implies the existence of a Borel include proper subspaces, spheres, lattices with positive doubling property of volume measures will be discussed 11 a) The set S satisfies -skew condition i.e. any and there gthe Property 27 turns turns out that that the the latter latter implies the former former while while the the former implies implies the existence existence of of ahold. Borel afor Borel ,lattices S out is closed and1implies null, .former Assume the following conditions 17 with positive codimension, cylinders, etc. Examples of remotely constant function 17 17 lattices lattices with with positive positive codimension, codimension, cylinders, cylinders, etc. etc. Examples Examples of ofthe remotely remotely constant function function Using this fact when is singleton, Grigor’yan and constant Saloff-Coste proved doubling proper volume measure of that space tothebe doubling. 17and lattices with positive codimension, cylinders, etc. Examples of constan 1 It Using this fact when isonly singleton, Grigor’yan andremotely Saloff-Coste pr Saksman, 28 measure 1998). which Therefore, is doubling(Luukkainen only doubling property & Saksman, 1998). Therefore, doubling property yer of the underlying spaces and not of the Dirichlet 28 measure measure which which is is doubling(Luukkainen doubling(Luukkainen & & Saksman, Saksman, 1998). 1998). Therefore, Therefore, only only doubling doubling property property 18 include positive rational functions, polynomial growth and subpolynomial growth functions such 18 18 include positive positive rational rational functions, functions, polynomial polynomial growth growth and and subpolynomial subpolynomial growth growth functions functions such such codimension, cylinders, etc. Examples of remotely 2 the weighted with finite unbounded weight functions. Moshimi and Tesei(Moshim while thehere. implies existence of Borel a) Theinclude set Sformer satisfies -skew condition i.e.measures forapositive any and there exists such that 18 include rational functions, polynomial growth and subpolynomial growth fun 2there weighted measures with finite unbounded weight functions. Mosh 12 b) The function is remotely constant i.e. exists a details, constant such that for any of volume measures will be discussed here. f.29 The book of Heinomen (Heinomen, 2001) is a on 29 of of volume volume measures measures will will be discussed discussed here. here. b) The function isexamples, remotely constant i.e. there exists a constant 12 19 as functions with logarithmic growth rates. For further details, examples, as well as methods to 19 19 as as functions functions with with logarithmic logarithmic growth growth rates. rates. For For further further details, examples, as as well well as as methods to to 3be Tesei, 2007) also used this fact to show doubling property ofmethods weighted Lebesgue measures & Saksman, 1998). Therefore, only doubling property 19 as functions with logarithmic growth rates. For further details, examples, well as constant function include positive rational functions, Definition 6. A non-zero Borel measure ν a metric 3 Tesei, 2007) also used this fact to show doubling property as of weighte 3013versions of doubling property, one is for the 30 ated 13 20 check whether weighted measure of this type satisfies doubling property, see (Tasena, 2011). 20 20Thecheck check whether whether aaaweighted weighted measure measure of of this this4type type satisfies satisfies doubling doubling property, property, see see (Tasena, (Tasena, 2011). 2011). 4 constant unique-singularity weight functions. To be precise, they proved doubling property for mea e. b) function is remotely i.e. there exists a constant such that for any 20 check whether a weighted measure of this type satisfies doubling property, see (Tasen unique-singularity functions. Toexists begrowth they proved doub polynomial growth and subpolynomial space is doubling doubling ifdoubling. there exists aametric on 3114 aDefinition metric Definition space Anon-zero non-zero is Borel ifmeasure measure there exists ametric metric space isdoubling doubling there exists 31 Definition 6.One 6. A6. A non-zero Borel measure on on aon aconstant space space isweight is doubling if there ififthere exists aprecise, a a functions Then is doubling only if there exists constant such that for and volume of that space toBorel be It 21 One interesting question isis to to classify weight functions for which their corresponding 21 21 measure One interesting interesting question question is toOne classify classify weight weight functions functions for for which which their their corresponding corresponding 14 is number doubling if and only ifdimension there exists constant such that 5if and where isinteresting aaThen positive less than the any of athan the underlying Euclidean su 21 question is to classify weight functions for which their cor 5 where is a positive number less the dimension of the d32 32 constant such that for any . of and a . growth rates. For further such as functions with logarithmic such that for any and constant such such that for for any any and and . . on aconstant metric space isthat doubling if there exists hile the former implies the existence a Borel 22 weighted measures satisfy doubling property. 22 22 weighted weighted measures measures satisfy satisfy doubling doubling property. property. 6 the fully accessible assumption was weaken to and that ofwas-skew condition and un 22thereweighted measures satisfy doubling is doubling if and only ifLater, exists a constant such that forproperty. any assumption 33Then 6 Later, thedetails, fully accessible to that of the -skew 33 3weaken 3 23 23 1998). Therefore, only 15 ksman, doubling nd23 . property examples, as well as methods to check whether a 23 15 3 7 singularity assumption was removed yielding the following result. ’yan 34 Grigor’yan 2005) and extended Saloff-Coste(Grigor’yan the concept &on Saloff-Coste, 2005) extended the concept of 34 Grigor’yan Grigor’yan andand Saloff-Coste(Grigor’yan Saloff-Coste(Grigor’yan && Saloff-Coste, Saloff-Coste, 2005) extended extended the the of of 7of singularity assumption was yielding the following. result. 24 Open Problem. Let be doubling measure on aaspaces metric space ..subsets Find necessary and 24 24& Saloff-Coste, Open Open Problem. Problem. Let Let be be doubling measure measure on aExamples metric metric space space .removed Find Find necessary necessary and and 16 Examples of -skew subsets of Euclidean include proper subspaces, spheres, 24aaadoubling Open Problem. Let be a 2005) doubling measure onconcept aconcept space Find nec 8 Saloff-Coste 16 of -skew ofmetric Euclidean spaces include pr weighted measure of this type satisfies doubling property, Grigor’yan and (Grigor’yan & Saloff8 anchored 35 doubling balls. They property showed to that that of doubling remote and property -anchored balls. They showed that doubling property 35 doubling doubling property property topositive to that that of of remote remote and -anchored -anchored balls. balls. They They showed showed that that doubling doubling property property or’yan &sufficient Saloff-Coste, 2005) extended theand concept of 17 lattices with codimension, cylinders, etc. Examples of remotely constant function 25 sufficient conditions of for which the weighted measure is doubling. 25 25 sufficient conditions conditions of of for for which which the the weighted weighted measure measure is is doubling. doubling. 9 Theorem 1.(Tasena, 2011) Let be a doubling measure on a metric space , 25 sufficient conditions of for which the weighted measure is do 17 lattices with positive codimension, cylinders, etc. Examples of rem Examples ofS-anchored -skew the subsets of Euclidean spaces include proper subspaces, spheres, 9with Theorem 1.(Tasena, 2011) Let comparison becomparison a doubling measure on a metric see (Tasena, 2011). Coste, 2005) concept ofpolynomial doubling property with 36 ofinclude remote property and S-anchored called volume balls comparison together with another property called volume comparison 36 of ofanother remote remote and and S-anchored balls balls together together with another another property property called called volume volume a-anchored metric space isextended doubling if there exists a balls. They showed that doubling property 18 positive rational functions, growth and subpolynomial growth functions such 26 26 26 26 18 include positive rational functions, polynomial growth and. subpolynom 10 , cylinders, S accessible. is 10 closed and null, and . function Assume the following conditions lattices with positive codimension, etc. Examples ofthe remotely constant ,that Sthat is closed and Assume theh balls provided condition that imply set doubling S property is fully for all balls provided that the set Sand is fully accessible. 3737 condition condition imply doubling doubling property property for for all all balls balls provided provided the set set S SPoincare is null, is fully fully accessible. accessible. One interesting question is to classify weight toimply that ofthe remote and S27 -anchored balls. They showed er with another property called volume comparison . 19 as functions with logarithmic growth rates. For further details, examples, as well as methods to 27 Poincare Inequality 27 27 Poincare Poincare Inequality Inequality Inequality 19 as functions with logarithmic growth rates. For further details, 11 a) The set S satisfies -skew condition i.e. for any condition and i.e. there existsand exampl suchthe t include positive rational functions, polynomial growth and subpolynomial growth functions such 11 a) Theisisis set S (uniform) satisfies -skew for anyweighted l balls provided that last the set S of28discussed is fullyand accessible. 20 check whether a weighted measure of this type satisfies doubling property, see (Tasena, 2011). 28 The last concepts discussed in this article the (uniform) Poincare inequality which 28 28 The The last concepts concepts discussed in in this this article article the the (uniform) Poincare Poincare inequality inequality which which S -anchored balls functions for which their corresponding that doubling property remote The last concepts discussed in this article is the (uniform) Poincare inequ 20 check whether a weighted measure of this type satisfies doubling proper as Saloff-Coste, functions with2005) logarithmic growth rates. For further details, examples, as well as methods to n 29 & extended concept of 21 One interesting question isvolume tohow classify weight functions for which corresponding 29 states how the -norm controlled by Dirichlet forms. Other inequalities of this type 29 states states how how the the -norm -norm isisisthe controlled controlled by by the the Dirichlet Dirichlet forms. forms. Other Other inequalities inequalities of of this this type typeOther 21 One interesting question is to classify weight functions for wo 29 states the -norm is controlled by the their Dirichlet forms. inequalities measures satisfy doubling property. together with another property called comparison b) The function constant i.e. there exists a constant 12 such for any check whether a weighted measure of this type satisfies property, see (Tasena, 2011). 12is remotely b) doubling The function isarticle remotely constant i.e. there exists a that constant hored They showed that doubling property 22 weighted measures satisfy doubling property. 30 include Sobolev inequality, Nash inequality, etc. Recent survey article by Saloff-Coste(Saloff30 30 balls. include include Sobolev Sobolev inequality, inequality, Nash Nash inequality, inequality, etc. etc. Recent Recent survey survey article by by Saloff-Coste(SaloffSaloff-Coste(Saloff30property Sobolev inequality, inequality, etc. Recent article weighted measures satisfy doubling 13 volume One interesting question is comparison toinclude classify weight functions forProblem. which their OpenNash Let condition imply doubling for all22 balls provided νcorresponding be aproperty. doublingsurvey measure on aby Saloff-Co ith31 another property called 13 23 23 31 Coste, 2011) discussed these concepts as well as their relationships. 31 Coste, Coste, 2011) 2011) discussed discussed these these concepts concepts as as well well as as their their relationships. relationships. 31 Coste, 2011) discussed these concepts as well as their relationships. weighted measures 14doubling Then is doubling and only ifmetric therespace exists abeconstant such athat any space andtha thatUnlike the set fullya property. accessible. Using this fact space necessary and sufficient 24 Open Problem. be doubling measure on a is metric .. Find Find necessary andfor s provided that the setsatisfy S Let isisproperty, fully accessible. 14 ifOpen Then doubling if and if there exists constant such 24 Problem. Let doubling measure on a metric 32 Unlike doubling property, the Poincare inequality property of Dirichlet forms. 32 32 Unlike doubling doubling property, the thePoincare Poincare inequality inequality isisisaaaproperty property of ofaonly Dirichlet Dirichlet forms. forms. 32 Unlike doubling property, the Poincare inequality is a property of Dirichlet for 25 sufficient of for the weighted measure isfor doubling. 33 33 33 when S conditions isLet singleton, Grigor’yan and Saloff-Coste proved conditions of for which the 25 which sufficient conditions of . Find necessary which the weighted weighted measure 33 Open Problem. be a doubling measure on a metric space and 34 Definition 7. Let be aa strongly strongly local regular Dirichlet form on with 26 34 34 Definition Definition 7. 7. of Let Let15 of weighted be bemeasures awhich strongly local local regular regular Dirichlet form form on on regular with with 34 Definition 7. Let be a strongly local Dirichlet form on 2615the sufficient conditions for measure isisdoubling. doubling property withweighted finite measureDirichlet doubling. Examples of -skew subsets of Euclidean spaces include proper subspaces, sphp 16 Examples of -skew subsets of Euclidean include 27 Poincare Inequality 35 associated energy measure . Then is said to satisfy weak (uniform) Poincare 35 35 associated associated energy energy16 measure measure . . Then Then is is said said to to satisfy satisfy weak weak (uniform) (uniform) Poincare Poincare 35 associated is said ofto Inequality satisfyspaces weak (uniform 27 energy measure . Then Poincare 17 lattices with positive codimension, cylinders, etc. Examples remotely constant fun unbounded weight functions. Moshimi and Tesei lattices with positive codimension, cylinders, etc. Examples of Pr 28 The last concepts in this article is the Poincare which 36 inequality for some constant and 36 36 inequality inequality ifififfor forsome some constant constant and and ,,fact , The 2817 last(uniform) concepts discussed insubpolynomial thisand article is growth the (uniform) 36discussed inequality if for some constant and ,inequality Poincare Inequality 1 this Using this when is singleton, Grigor’yan Saloff-Coste proved dou positive rational functions, polynomial growth and functions (Moshimi & Tesei,18 2007)include also used fact to show 18 include positive rational functions, polynomial growth and subpolyno Poincare Inequality 29 states how the -norm is controlled by the Dirichlet forms. Other inequalities of this type 29 states how the -norm is controlled by the Dirichlet forms. Other The last concepts19discussed in2thiswith article is measures the (uniform) Poincare inequality which weighted withrates. finite For unbounded weight functions.as Moshimi and as functions growth further details, examples, well as metho 19logarithmic aswith functions with logarithmic rates. For further details, exam 30 include Nash inequality, etc. Recent survey article by growth Saloff-Coste(Saloffdoubling property of controlled weighted Lebesgue measures 30 include Sobolev inequality, Nash inequality, etc. Recent survey article The last concepts discussed in this article is the states how theSobolev -norminequality, is by the Dirichlet forms. Other inequalities of this type 3 Tesei, 2007) also used this fact to show doubling property of weighted Lebesg 7 a20 20 checkconcepts whether weighted measure ofathis type satisfies doubling property, seedoubling (Tasena, prop 2011 201 7this type check whether weighted measure ofas satisfies 31 Coste, 2011) discussed these as well as their relationships. 31 Coste, 2011) discussed concepts wellstates asinequality. their relationships. unique-singularity weight functions. Tointeresting be precise, they 37 If then isis said to satisfy (uniform) Poincare inequality. 37 37 If IfSobolev ,,,then then is said said to satisfy (uniform) (uniform) Poincare Poincare inequality. inequality. include inequality, Nash inequality, Recent survey article bythese Saloff-Coste(Saloff(uniform) Poincare inequality which how the 4 satisfy unique-singularity functions. To befunctions precise, they proved doubling prop 37 Ifto ,etc. then isweight said toclassify satisfy (uniform) Poincare 21 One question is to weight for which their correspon 21 this One question is Poincare to classify weight is functions for 32 Unlike doubling property, the Poincare inequality isinteresting a property of Dirichlet forms. 7inequality 1well Using fact when is singleton, Grigor’yan and Saloff-Coste proved dou 32 Unlike doubling property, the a property o 38 38 38 2 a positive Coste, 2011) discussed these concepts as as their relationships. 38 proved doubling property for measures where 5 where is number less than the dimension of the underlying 22 weighted measures satisfy doubling property. Lmeasures -norm isfinite controlled by theweight Dirichletfunctions. forms. Other 33 22 weighted satisfy doubling property. 33 2 weighted measures with unbounded Moshimi and 7Poincare and Saloff-Coste proved doubling property 39 Under doubling property, weak Poincare inequality and Poincare inequality are 39 39 Under Under doubling doubling property, property, weak weak Poincare inequality inequality and and Poincare Poincare inequality inequality are areand ng this fact when is Grigor’yan and proved doubling property of 23 Unlike property, the inequality aaccessible property of Dirichlet forms. 7Poincare 39 Under doubling property, weak inequality Poincare ineq 6theaof Later, the fully assumption was weaken to local that ofregular -skew conditio 23Saloff-Coste λ is adoubling positive number less thanPoincare dimension oflocal theisalso 34 Definition 7. singleton, Let be strongly regular Dirichlet form on with inequalities of this type include Sobolev inequality, 34 Definition 7. Let be a strongly Dirichlet 3 Tesei, 2007) used this fact to show doubling property of. weighted Lebesg 3 ht functions. Moshimi and Tesei(Moshimi & 40 equivalent (Saloff-Coste, 2002). 40 40 equivalent equivalent (Saloff-Coste, (Saloff-Coste, 2002). 2002). 24 Open Problem. Let be a doubling measure on a metric space Find necessary ghted measures with finite unbounded weight functions. Moshimi and Tesei(Moshimi & 40 equivalent (Saloff-Coste, 2002). 7 singularity assumption was removed yielding the following result. 2477accessible Open Let be a . doubling measure on asaid metric space w 1 35 Using this fact when measure isspace. singleton, Grigor’yan and Saloff-Coste proved doubling property of associated energy . strongly Then isProblem. said to satisfy weak (uniform) Poincare the property fully 35 energy measure Then is Salofftois satisfy 7 Recent Nash inequality, etc. survey article 4conditions unique-singularity weight functions. To be precise, they by proved doubling prop Definition 7. Let be doubling aLater, localassociated regular Dirichlet formmeasures on with gei,property ofunderlying weighted Lebesgue measures 8with 2007) also used thisEuclidean fact of weighted Lebesgue with 25to show sufficient of for which the weighted measure doubling 25 sufficient conditions of for which the weighted measure 2 weighted measures with finite unbounded weight functions. Moshimi and Tesei(Moshimi & 36 inequality if for some constant and , ’yan and Saloff-Coste proved doubling property of 36 inequality if for some constant and , ct associated when isassumption singleton, Grigor’yan and Saloff-Coste doubling property of condition and doubling was weaken to that ofprecise, energy measure isproved said to satisfy weak (uniform) Poincare where is a2011) positive than the dimension the underlying 9k5-skew Theorem 1.(Tasena, Letnumber be a less doubling measure on aofmetric space ise, they proved doubling property forThen measures que-singularity weight functions. To be they property for measures Coste (Saloff-Coste, 2011) discussed these 26this .fact 26& proved 3 Tesei, 2007) also used to show doubling property of weighted Lebesgue measures with concepts as weight functions. Moshimi and Tesei(Moshimi asures with finite unbounded weight functions. Moshimi and Tesei(Moshimi & 3 3 3 inequality if for some constant and , 6theremoved Later, the fully accessible assumption was weaken to that of -skew conditio 3yielding 27that Poincare Inequality he of the underlying Euclidean space. 10 isthe closed null, .Inequality Assume the following assumption was where isunique-singularity a positive number less than of underlying Euclidean space. Note that fully accessibility is equivalent to -skew condition. Note Note that fully fully accessibility accessibility isdimension isequivalent equivalent to tothat -skew -skew condition. condition. well asand their relationships. ’yan yan and and Saloff-Coste Saloff-Coste proved proved doubling doubling property property of of , Sproved 27 Poincare Note fully accessibility isand equivalent to measures -skew condition. 4dimension unique-singularity weight functions. Toof be precise, they proved doubling property for ct when isthe singleton, Grigor’yan and Saloff-Coste doubling property of 3 bling property of weighted Lebesgue measures with also used this fact to show doubling property weighted Lebesgue measures with 3 7 singularity assumption was removed yielding the following result. 28 The last concepts discussed in this article is the (uniform) Poincare inequality en5 the to that ofthe -skew condition and the uniqueer, fully accessible assumption was weaken to that -skew condition and & the unique11 lessa)than The setof S satisfies -skew condition i.e. for any and existsw weight weight functions. functions. Moshimi Moshimi and and Tesei(Moshimi Tesei(Moshimi & & following 28 The last concepts discussed in this article is the there (uniform) asures with finite unbounded weight functions. Moshimi and Tesei(Moshimi Unlike doubling property, the(uniform) Poincare inequality where isresult. a positive number the dimension of the underlying Euclidean space. precise, they proved doubling property for measures 8 3, then 3then 37 If is said to satisfy (uniform) Poincare inequality. arity weight functions. To be precise, they proved doubling property for measures 37 If , is said to satisfy Poincare inequality 29 states the -norm isthat controlled by condition thewith Dirichlet forms. Other inequalities of Othe this following result. gularity assumption removed yielding following result. bling bling property property of offully weighted weighted Lebesgue Lebesgue measures measures with with 29 states how the isbecontrolled bymeasure the Dirichlet forms.space also used this fact towas show doubling property of weighted Lebesgue measures 6 38 Later, the accessible assumption weaken to ofaEuclidean -skew the uniqueTheorem 1. (Tasena, 2011) Let νhow bethe doubling measure 9awas Theorem 1.(Tasena, 2011)-norm Let aand doubling on a metric han the dimension of the underlying Euclidean space. is property of Dirichlet forms. re is a positive number less than the dimension of the underlying space. 38 30 include Sobolev inequality, Nash inequality, etc. Recent survey article by Saloff-Coste(Sa 3 inequality, , then is said to satisfy (uniform) Poincare inequality. 12measures b) 30 The function is remotely constantNash i.e. there exists aetc. constant such precise, precise, they they proved proved doubling doubling property property for for measures Sobolev inequality, Recentthe survey artit arity To bewas precise, they proved doubling property for measures 7If singularity assumption removed yielding the result. weaken to -skew condition and the uniqueon aofUnder metric space 39 weight doubling weak Poincare inequality and inequality are 10 , following Sinclude isconcepts closed and null, and . Assume following measure onthat afunctions. metric space eorem 1.(Tasena, 2011) Let beCoste, a, property, doubling measure on a metric space ,Poincare Definition 7. Let (E,D(E)) be a strongly local regular ly accessible assumption was weaken to that of -skew condition and the unique39 Under doubling property, weak Poincare inequality and 31 2011) discussed these as well as their relationships. 13 8 3 underlying an an the the dimension of of(Saloff-Coste, the the underlying Euclidean Euclidean space. space. 31the Coste, 2011) discussed these2 concepts as well asand their relationships. 3 underlying e the adimension number less than the dimension of Euclidean space. gsumption result. equivalent 2002). 11a doubling a) The set S satisfies -skew condition i.e. for there exists .positive Assume the following conditions hold. was removed yielding the following result. 40 (Saloff-Coste, 2002). ,40 Sisfollowing isisclosed and null, and ..equivalent Assume the following conditions hold. closed and null, and Assume 32 Unlike doubling property, the Poincare inequality is aany property of Dirichlet forms. Dirichlet form on L (X, ν ) with associated energy Under doubling property, weak Poincare inequality and Poincare inequality are 9 Theorem 1.(Tasena, 2011) Let be measure on a metric space , 14 Then is doubling if and only if there exists a constant suchisthat for any weaken weaken to to that that of of -skew -skew condition condition and and the the uniqueunique32 Unlike doubling property, the Poincare inequality a property ly accessible assumption was weaken to that of -skew condition and the unique33 3 exists such The set Sand satisfies -skew condition i.e. for any 33 3and there exists such that is said equivalent 2002). the following hold. 10 , (Saloff-Coste, Sremoved is3be closed and null, and . Assume the conditions hold. measure Γ .following Thenlocal (E,D(E)) to satisfy weakon gysumption the the following following result. result. bling measure on athere metric space , that was yielding following Tasena, 2011) Let aconditions doubling measure on aLet metric space 34 the Definition 7.result. be a ,Let strongly regular Dirichlet form b) The function is remotely constant i.e. there exists a local constant 12 such t 34 Definition 7. be a strongly regular Dirichle 15 a) The set S satisfies k -skew condition i.e. for any 11 a) The set S satisfies -skew condition i.e. for any and there exists such that . Assume the conditions hold. (uniform) Poincare inequality if for some constant 3 following is closed and null, and . Assume the following conditions hold. 3 35that associated energy measure . ofenergy Then isof satisfy weak (uniform) Poin thatfor fully accessibility is equivalent to -skew condition. exists a constant such any ling ing function measure measure on on a metric metric space space ,there , 1613 exists The is constant i.e. a constant such that for accessibility any associated measure . said Then is said toproper satisfy Note fully istoequivalent to -skew condition. Examples -skew subsets Euclidean spaces include su Tasena, Letaremotely bethere aNote doubling measure onthat a35 metric space , that r any 2011) and exists such that and there exists such and atisfies -skew condition i.e. for any and there exists such that 36 inequality if for some constant and , 3 14 Then is doubling if and only if there exists a constant such that for any 36 inequality if for some constant and , . . Assume Assume the the following following conditions conditions hold. hold. 17 lattices with positive codimension, cylinders, etc. Examples of remotely fully accessibility is .equivalent -skew is closed andfunction null,Note andthat Assume the following conditions hold. b) The is remotely constant i.e. thereto exists a condition. constant 12 such that for any nstant such that for any and 18 include positive rational functions, growth and subpolynomial grow nere is doubling if and only if there exists a constant such that for any and ratisfies any any and and there there exists exists such such that that -skew condition i.e. for any and there such existsthat for such that polynomial 13 exists a constant that foraany on is remotely there exists constant any growth 15 b) Theconstant functioni.e. a issuch remotely constant i.e. there existswith a logarithmic 19 as functions rates. For details, examples, as w 14 Then is doubling if and only if there exists a constant such-skew that for any of Euclidean and further 16 Examples of subsets spaces include properseesu 37 If , then is said to satisfy (uniform) Poincare inequality. check a weighted measureisofsaid thistotype satisfies doubling property, constant such that for any ere ere exists exists aa constant constant such such that that for for20 any 37 whether If such ,________________________________________________________ thenfor any satisfy (uniform) Poincare inequali on is remotely constant i.e.for there exists aany constant that a constant such that any and 17 lattices with positive codimension, cylinders, etc. Examples of remotely bling if andinclude only exists a constant such that for any3 Note and ean spaces proper subspaces, spheres, Examples of if there -skew subsets of Euclidean spaces proper subspaces, 38 21 Oneinclude interesting question is to spheres, classify functions that fully accessibility is equivalentweight to 1‐skew condition. for which th 38 15 18 etc. include positive rational functions, polynomial growth subpolynomial grow etc. Examples ofsuch remotely constant function ces with positive codimension, cylinders, ofUnder remotely constant function 39-skew Under doubling property, weak Poincare inequality andandPoincare inequality 22 measures satisfy doubling property. 39Examples doubling property, weak Poincare inequality and aabling constant constant such that that for for any any and and 16 Examples of subsets of weighted Euclidean spaces include proper subspaces, spheres, if and only if there exists a constant such that for any and 19 as functions with logarithmic growth rates. For further details, examples, as w 23 rowth and subpolynomial growth functions such ude positive rational functions, polynomial growth and subpolynomial growth functions such 40 equivalent (Saloff-Coste, 2002). 40 etc. equivalent (Saloff-Coste, 2002). 17 with positive codimension, cylinders, Examples of remotely function doubling property, see uclidean spaces include proper subspaces, ples oflattices -skew subsets of Euclidean spaces include proper 20 spheres, check whether weighted ofconstant this type 24 Open Problem. Let be awell doubling measure on a metric space . F further details, examples, asgrowth well asrates. methods to unctions with logarithmic For further details, examples, as measure as methods to satisfies Santi Tasena /growth Burapha Sci.asubspaces, J. 19 (2014) 1spheres, : 169-177 174 18 include positive rational functions, polynomial and subpolynomial growth functions such functions for which th ers, etc. Examples of remotely constant function positive codimension, cylinders, etc. Examples of remotely constant function 21 One interesting question is to classify weight 25 sufficient conditions of for which the weighted measure tisfies doubling property, see (Tasena, 2011). ck a weighted measure of this type satisfies doubling property, see (Tasena, 2011). clidean clidean spaces include include proper proper subspaces, subspaces, spheres, spheres, 19 whether functions with logarithmic growth rates. For further details, examples, as well as methods to ples ofasspaces -skew subsets of Euclidean spaces include proper subspaces, spheres, ial growth and subpolynomial growth functions such ve rational functions, polynomial growth and subpolynomial growth functions such 22 weight weighted measures satisfy doubling property. 33 26 weight functions for which their corresponding One interesting question is to classify functions for which their corresponding 3property, Note Note that that fully fully accessibility accessibility is is equivalent equivalent to to -skew -skew condition. condition. ers, rs, etc. etc. Examples Examples of of remotely remotely constant constant function function 20 check whether a weighted measure of this type satisfies doubling see (Tasena, 2011). positive codimension, cylinders, etc. Examples of remotely constant function For further details, examples, as well as methods to 23 Note that fully accessibility is equivalent to -skew condition. 17 is path-connected. 17 is19 path-connected. Theorem 2.(Tasena, 2011) Let be a strongly loca 18 18 with strongly local regular Dirichlet form on 7 is19path-connected. 20 with associated energy be alocal k-accessibl Theorem 2.(Tasena, 2011) Let be a strongly local regular Dirichlet form on Theorem 2.(Tasena, 2011) Let measure be, a strongly regular n8 is said to satisfy weak (uniform) 19 Poincare 21 . Define 20 with associated energy measure , be a k-accessible null set, , and 20 with associated energy measure , be a k-accessible null se 9 Theorem 2.(Tasena, 2011) Let be a strongly local regular Dirichlet form on and , 21 . Define 21 a k-accessible . Define 0 with associated energy measure , be null set, , and for any 22 for any . Then 1 . Define 23 regular Dirichlet form , called the weighted Diri Iffork =any1, then (E,D(E)) is said to satisfy Then extended toto aastrongly local regular 22 . Then extended strongly 22 (uniform) ext for any . local Then sfy (uniform) Poincare inequality. 24 Moreover, if satisfies Poincare inequality, 23 regular Dirichlet form , called the weighted Dirichlet form , on . Dirichletextended form called theweighted weightedDirichlet for 23 . regular form , ,called the 2 for anyPoincare inequality. to a strongly local Then Dirichlet 25 is a nonincreasing remotely constant function, then satisfie 24 Moreover, if satisfies Poincare inequality, satisfies doubling property, and 24 weighted Moreover, if ,,on satisfies Poincare inequality, Dirichlet Underform doubling property, weak Poincare Dirichlet form form 3 regular , called the on . weak Poincare inequality and Poincare inequality are Dirichlet 26 and only if the measure is doubling. 25 is a nonincreasing constant function, satisfies Poincare inequality if 25 is a then nonincreasing remotely constant function, sati 4 Moreover, ifPoincareremotely satisfies Poincare inequality, doubling property, and satisfies Poincarethen inequality, inequality and inequality are equivalent (Saloff satisfies Moreover, if (E,D(E)) 27 26 and only if the measure is doubling. 26 then and only if the measure is doubling. 5 is a nonincreasing satisfies Coste, 2002). remotely constant function, ν satisfies doublingPoincare property,inequality and a is a ifnonincreasing 28 Sturm’s Result 27 27 6 and only if the measure is doubling. h 1996) h 29 Sturm(Sturm, 1994, 1995b, showed in a series Grigor’yan and Saloff-Coste (Grigor’yan & Saloff- Result remotely constant function, then (ESturm’s ,D(E ))Result satisfies 28 28 Sturm’s y7is equivalent to -skew condition. 30 of Sturm(Sturm, parabolic Harnack Poincare inequality, 29 Sturm(Sturm, 1994, 1995b, 1996) in a seriesinequality of articles that estimates, Coste, 2005) extended Poincare inequality that Poincare if1995b, and Gaussian only if the measure is andofdoubli 29 to showed 1994,inequality, 1996) showed in aµ series articl 8 Sturm’s Result 31 30 parabolic Harnack inequality, Poincare inequality, and doubling property are closely related. Harnack inequality, Poincare inequality, be anda doubling balls, where ⊂30X is aparabolic fully remote and S-anchored doubling. 9 Sturm(Sturm, 1994, 1995b, 1996)S showed in32 a series of articles that Gaussian estimates, Theorem 3.(Sturm, 1996) Let strongly prope local 31 31 0 parabolic Harnack inequality, Poincare inequality, and doubling property are closely related. 33 Then the following statements are equivalent. accessible set. They used this fact to prove Poincare 32 Theorem 3.(Sturm, 1996) Let a strongly local regular form on . local regular 32 beTheorem 3.(Sturm, 1996)Dirichlet Let be a strongly 1 34 a) The Dirichlet form satisfies parabolic Harnack 8 8 33 Then the following statements are equivalent. inequality on weighted when S is alocal Then theSturm’s following statementsform are equivalent. 2 Theorem 3.(Sturm, 1996) LetDirichlet spaces be 33 a strongly regularResult Dirichlet on . 35 b) The Dirichlet form satisfies Gaussian estimates 34 the a)following The Dirichlet Harnackform inequality. 34 parabolic The Dirichlet parabolic Harnack inequalit singleton andstatements the form geodesic Xsatisfies satisfies aa)mild 3 Then arespace equivalent. The Sturm (Sturm, 1994, satisfies 1995b, 1996) showed in a inequality 36 c) Dirichlet form satisfies Poincare 35 b) The Dirichlet form satisfies Gaussian estimates. 35 b)Harnack The Dirichlet form satisfies Gaussian estimates. 4 and a) The Dirichlet form satisfies parabolic inequality. nand Saloff-Coste(Grigor’yan Saloff-Coste(Grigor’yan && Saloff-Coste, Saloff-Coste, 2005) 2005) extended extended Poincare additional assumption called relatively connected series ofPoincare articles that Gaussian estimates, parabolic 37 doubling property. 36 c) The Dirichlet form satisfies Poincare inequality and the volume measure satisfies 36 c) The Dirichlet form satisfies Poincare inequality and the b) The Dirichlet form satisfies Gaussian estimates. f5of remote remote andand -anchored -anchored balls, balls, where where is is a fully a fully accessible set. set. They They 38accessible annuli. Their weight functions include strictly positive Harnack inequality, Poincare inequality, and doubling 37 doubling property. 37 doubling 6 Poincare c)38 The Dirichlet form satisfies Poincare inequality the volume measure ove prove Poincare inequality inequality on on weighted weighted Dirichlet Dirichlet spaces spaces when when isproperty. aisand singleton a singleton 39 The fact 8that a) and satisfies b) are equivalent is perhaps 38singularity. property are closely related. unbounded real-valued functions with no 7space doubling property. 40 equivalent to c). Since solutions heatisare equations are inte pace satisfies satisfies a mild a mild additional additional assumption assumption called called relatively relatively connected connected 39 The fact that a) and b) are 39 equivalent isThe perhaps less a)surprising thanequivalent thatofthey fact that and b) are perhaps less sur 8 Later, Moshimi and Tesei(Moshimi & Tesei, 2007) extended Theorem 3. (Sturm, 1996) Let (E,D(E)) be a strongly 41 functions heat kernels the initial heat profiles, the are behavior of hea ht ight functions functions include include strictly strictly positive positive unbounded unbounded real-valued real-valued functions with with noand noofsolutions 40 equivalent to c). Since solutions of heat equations are integrals the multiplication between 40 equivalent to c). Since of heat equations integrals of yan and Saloff-Coste(Grigor’yan & Saloff-Coste, 2005) extended Poincare 9 The fact and thatSaloff-Costes’ a) and b) are equivalent is perhaps less surprising than that they are 2 Grigor’yan work to include weight 42 behavior of solutions of heat equations. On the contrary, hea Moshimi ,atMoshimi and and Tesei(Moshimi Tesei(Moshimi & & Tesei, Tesei, 2007) 2007) extended extended Grigor’yan Grigor’yan and and SaloffSalofflocal regular Dirichlet form on L (X, ν ) . Then the 41 heat kernels and the initial heat profiles, the behavior of heat kernels will directly influence the 41is a heat andof thethe initial heat the behavior of heat kernels remote andto c). -anchored balls, where fully accessible set. They 0 of equivalent Since solutions of heat equations arekernels integrals multiplication between 88 profiles, functions where nclude weight weight functions functions where where 42 spaces . when Nevertheless, .contrary, Nevertheless, the the 42 behavior of solutions of heat equations. On the heat kernels can be viewed solutions following statements are equivalent. behavior of solutions of heat equations. On the contrary, heat kernels olude prove Poincare inequality on weighted Dirichlet is a singleton 1 heat kernels and the initial heat profiles, the behavior of heat kernels will directly influenceasthe an be be applied applied to other to other weight weight functions functions with with unique unique singularity. singularity. 4 8 Nevertheless, the same technique can be applied to (E,D(E)) satisfies parabolic and relatively a) The Dirichlet form sic a mild additional assumption called relatively connected 2 space behaviorsatisfies of solutions of heat equations. On the contrary, heat kernels can In be viewed as solutions case of singleton , fully accessibility Tasena(Tasena, sena(Tasena, 2011) extended extended these these results toSaloff-Coste, to include include weight weight functions functions with with 8results gor’yan and 2011) Saloff-Coste(Grigor’yan & Saloff-Coste, 2005) extended Poincare or’yanfunctions and Saloff-Coste(Grigor’yan &unbounded 2005) extended Poincare 4 accessibility. The converse is not true, however. weight include strictly positive real-valued functions with no other weight functions with unique singularity. Harnack inequality. In case of singleton fully accessibility that andthat relatively connected annuli together implies - and relatively connected In caseset. of singleton accessibility es. ities. Heof He proved Poincare Poincare inequality inequality under under the,the assumptions assumptions thethe weight weight that ofproved remote and -anchored balls, where isaaand fullySaloff-Coste(Grigor’yan accessible They ,&fullySaloff-Coste, oto that remote and -anchored balls, where is fully accessible set. They er, Moshimi and Tesei(Moshimi & Tesei, 2007) extended Grigor’yan and Saloff1 Grigor’yan 2005) extended accessibility. The converse is not true, however. 2011) extended these 42011, Tasena (Tasena, The converse is not and Poincare Saloff-Coste(Grigor’yan 2005) Poincare b)extended The accessibility. Dirichlet form (E,D(E)) satisfies Gaussian eor’yan the form form ,Inwhere , In where is the the singularity singularity setset andand is is a when nonincreasing awhen nonincreasing case ofissingleton ,& fullySaloff-Coste, accessibility and relatively connected annuli together implies - true, however. act to prove inequality on weighted Dirichlet spaces is a singleton ct to prove Poincare inequality on weighted Dirichlet spaces is a singleton off-Coste, include weight functions where . accessible Nevertheless, the 2balls, inequality to that of remote and -anchored is a fully accessible 2005) extended Poincare oeodesic that ofspace remote and -anchored where isadditional aadditional fully set. Theyballs, where results toaccessibility. include weight functions with multiple The converse issatisfy not true,anhowever. estimates. unction. tcan function. The The singularity singularity set must must also also satisfy an assumption assumption satisfies aset mild additional assumption called relatively connected odesic space satisfies a mild additional assumption called relatively connected be applied to other weight functions with unique singularity. 3 used this fact to prove Poincare inequality on weighted Dirichlet spaces when is a 4to 4prove is a fully accessible set. ct Poincare inequality on weighted Dirichlet spaces when isDirichlet a singleton ility. y. This This is ais stronger a stronger version version of They of-skew -skew condition condition mentioned mentioned previously. previously. singularities. He proved Poincare inequality under the eir weight functions include strictly positive unbounded real-valued functions with no c) The form (E,D(E)) satisfies Poincare weight functions include strictly positive unbounded real-valued functions with no ,ir Tasena(Tasena, 2011) extended these results to include weight functions with 4 and the geodesic called space relatively satisfiesconnected a mild additional assumption called relatively richlet spaces when is aa mild singleton desic satisfies additional assumption .Later, Later,space Moshimi andTesei(Moshimi Tesei(Moshimi & Tesei, 2007) extended Grigor’yan andthe Saloffassumptions that the weight functions are of the form Moshimi and & Tesei, 2007) extended Grigor’yan and Saloffν satisfies doubling inequality and volume measure arities. He proved Poincare inequality under the assumptions that the weight 5 annuli. Their weight functions include strictly unbounded real-valued function umption called connected losed edweight subset subset of of a relatively geodesic a geodesic space positive is said is unbounded said to to be be-accessible -accessible if itfunctions if satisfies it satisfies ir functions includespace strictly real-valued with no positive ork toinclude include weightfunctions functions where . Nevertheless, the k to weight where . Nevertheless, the f the form , where is the singularity set and is a nonincreasing , where S is the singularity set and a is a singularity. Later, Moshimi and Tesei(Moshimi property. ounded real-valued functions no& nor for some some and and for for anyany with6and and , the set set Later, Moshimi and Tesei(Moshimi Tesei,, the 2007) extended Grigor’yan and Saloff- & Tesei, 2007) extended Grigor’yan a nique iquefunction. can beapplied applied to otherweight weight functions with unique singularity. que can be to other functions with unique singularity. ant The singularity set must also satisfy an additional assumption 7 Costes’ work to include weight functions Grigor’yan and Saloffnonincreasing remotely constant function.where The singularity . Nevertheless, The fact that a) and b) arewhere equivalent is . Neverth k07) to extended include weight functions the 4 2011, Tasena(Tasena, 2011) extended these results to include weight functions with 011, Tasena(Tasena, 2011) extended these results to include weight functions with ibility. This is a stronger version of -skew condition mentioned previously. 8the same technique can be applied to other weight functions with unique singularity. . can be applied where . toNevertheless, que other weightan functions with unique singularity. must also satisfy additional assumption called perhaps lessthe surprising than that they are equivalent to ngularities. set HeSproved proved Poincare inequality under theTasena(Tasena, assumptions that the weight ngularities. He Poincare inequality under the assumptions that weight 9 In 2011, 2011) extended these results to include weight func 4 extended these results to include weight functions hclosed unique singularity. 011, Tasena(Tasena, 2011) with k-accessibility. is10a stronger -skew subset of a geodesic space isversion said Dirichlet toofDirichlet bekset -accessible ifanonincreasing itsolutions satisfiesof heat equations are integrals of the are the form ,This where is the singularity set and is nonincreasing c). Since ofof the form , where is the singularity and is a ena, ,elts2011) 2011) Let Let be be a strongly a strongly local local regular regular form form on on multiple singularities. He proved Poincare to include functions with gularities. Heweight proved Poincare inequality under the that the weightinequality under the assumptions that th on for some and for any andmust , the setassumptions onstant function. The singularity setfunctions must also satisfy anadditional additional assumption condition mentioned previously. nstant function. The singularity set also satisfy an assumption multiplication between is heatthekernels and the energy rgy measure measure , , be be a k-accessible a k-accessible null null set, set, , and , and 11 are of the form , where singularity set initial and is a noni der weightis the singularity set and is a nonincreasing ecessibility. ofthe theassumptions form , the where 4 44 This is athat stronger version of -skew condition mentioned previously. essibility. This is a stronger version of -skew condition mentioned previously. Definition 8. A closed subset S of a geodesic space (X,d) e 12set remotely constant singularity set of must also satisfy an additional as heat The profiles, the behavior heat kernels will directly ularity set and The is a singularity nonincreasing nstant function. must also satisfyfunction. an additional assumption ed. 4 4 13 called -accessibility. This is a stronger version of -skew condition mentioned previo issubset said be ifspace itofsatisfies k’-skew condition lso an additional assumption essibility. This istoa of stronger version -skew mentioned previously. influence the of solutions of heat equations. On 8.AAsatisfy closedsubset ofaka-accessible geodesic issaid said be -accessible itsatisfies satisfies 8. closed geodesic space iscondition totobe -accessible ififitbehavior 14 . for Then Then local extended toset a strongly a the strongly local condition mentioned asena, 2011) Letsome kpreviously. strongly regular Dirichlet form onlocal for ’>and kand andbe fora.any any and extended , the the ndition forsome some any and setto dition for for and set contrary, heat kernels can be viewed as solutions of 15 Definition 8. ,A,the closed subset ofifa itgeodesic space is said to be -accessible if . A closed subset of a geodesic space is said to be -accessible satisfies form , called , called weighted weighted Dirichlet Dirichlet form form ,set, on, on . . drm energy measure , thethe be a k-accessible null , and heat equations Dirac measures 16 -skew condition for some and with for any andas initial ,heat the profiles, set s, ifsaid to -accessible if itforsatisfies dition forbe some and any and ,doubling thedoubling set satisfies satisfies Poincare Poincare inequality, inequality, satisfies satisfies property, property, andand nected. fine ected. is path-connected. , the set soinequality parabolic Harnack remotely g remotely constant constant function, function, then then satisfies satisfies Poincare Poincare inequality if if inequality will force the heat kernels 17 is path-connected. (E,D(E)) be a strongly Theorem 2. (Tasena, 2011) Let ected. 2.(Tasena, 2011) Let be a stronglylocal localregular regularDirichlet Dirichlet form on estimates. The Poincare inequality and Gaussian (Tasena, Let be form on easure ure is doubling. is2011) doubling. 18 . a strongly Then2 extended to to a satisfy strongly local iatedenergy energylocal measure , beaon ak-accessible k-accessible nullset, set,2011) Let ,and and regular ,Dirichlet L (X, ν) withnull associated ated measure be , be 19 Theorem a strongly local regular Dirichlet form on ________________________________________________________ et form 2011) thebeform weighted Dirichlet form , on . on (Tasena, Let , called a strongly local2.(Tasena, regular Dirichlet form Define Sturm’s Sturm’s Result Result 4 Define energy measure ⊂ inequality, X a k-accessible set, measure 20 with associated energy , and,S,and be accessibility a k-accessible null connected set, al regular form onΓ , S be er, ifenergyDirichlet satisfies satisfies doubling Inproperty, case of singleton fully and relatively ated measure , Poincare a be k-accessible nullnull set, m,rm, 1994, 1994, 1995b, 1995b, 1996) 1996) showed showed in21and in a series a series of of articles articles that that Gaussian Gaussian estimates, estimates, .. Define Define annuli together implies le null set, constant function, ,then and ing remotely satisfies Poincare inequality if k-accessibility. The converse is not true, nequality, kDefine inequality, Poincare Poincare inequality, inequality, andand doubling doubling property property are are closely closely related. related. . Then extended to a strongly local . Then extended to a strongly local measure is doubling. however. richlet form , called the weighted Dirichlet form , on . chlet form the Dirichlet form form , on . . . 22 weighted for any . Then extended to a stron m, 1996) 1996) LetLet be,be acalled strongly a strongly local regular regular Dirichlet Dirichlet form on on .local Then extended to a strongly local reover,ifif satisfies Poincare inequality, satisfies doubling property, and eover, satisfies Poincare inequality, satisfies doubling property, and Sturm’s 23Result regularDirichlet Dirichletform form, on , called the weighted Dirichlet form , on . extended to a strongly local gtatements statements areare equivalent. equivalent. chlet form , called the weighted .(2557) สั น ติ ทาเสนา / วารสารวิ ท ยาศาสตร์ บ ร ู พา. 19 1 : 169-177 175 creasing reasing remotely constant function, then satisfies Poincare inequality if easing remotely constant function, then satisfies Poincare inequality if Sturm, 1994, 1995b, 1996) showed in a series of articles that Gaussian estimates, 24Harnack Moreover, if doubling property, satisfies Poincare inequality, satisfies doubling prope ichlet ifform , satisfies onsatisfies . Poincare m orm parabolic parabolic Harnack inequality. inequality. eover, satisfies inequality, satisfies and themeasure measure isPoincare isdoubling. doubling. he ack inequality, inequality, and doubling property are closely related. 25 is a nonincreasing remotely constant function, then satisfies Poincare in ,m satisfies doubling property, and orm remotely satisfies satisfies Gaussian Gaussian estimates. estimates. easing constant function, then satisfies Poincare inequality if 26inequality only if the the measure is doubling. m orm satisfies satisfies satisfies Poincare Poincare inequality and the volume volume measure measure satisfies satisfies Poincare inequality if and and 4 the doubling property, on the other hand, have almost the weighted Dirichlet spaces satisfy the parabolic 99 nothing to do with heat equations. Yet these properties Harnack inequality. Even whether k-accessibility is the directly influence both heat Harnack kernels and solutions of right condition to assume on singularity sets is unknown. s asas initial initial heat heat profiles, profiles, so so parabolic parabolicHarnack inequality inequality 9 9 heat equations. AsPoincare an example, Tasena’s result can be All these require further investigation into the subject 9 Gaussian Gaussian estimates. estimates. The The Poincare inequality inequality and and the the 9 translated asdo follows. so the author strongly believes this field will remain have ave almost almostnothing nothingtoto do with withheat heatequations. equations. Yet Yetthese these ac Dirac measures measures as1initial as initial heat heat profiles, profiles, sowith so Harnack Harnack kernels kernels and andsolutions solutions heat heat equations. equations. As Asparabolic an anexample, example, of4.ofof heat equations Dirac measures as inequality initial heat profiles, Theorem Let (E,D(E)) be aparabolic strongly local regularinequality fruitful in years to come.so parabolic Harnack inequality rnels s to satisfy to satisfy Gaussian Gaussian estimates. estimates. The The Poincare Poincare inequality inequality and and the the ows. ows. 2 will kernels to satisfy Harnack inequality Dirichlet form force on L2the (X, νheat ) with associated energyGaussian estimates. The Poincare inequality and the other he other hand, hand, havehave almost almost nothing nothing to do to with do heatother heat equations. equations. YetYet these these 3the doubling property, onwith the hand, have almost nothing to do with heat equations. Yet these inequality and measure Γ and , S solutions ⊂on X isof aheat ρ -accessible null set, y y local local regular regular Dirichlet Dirichlet form form on with with associated associated Acknowledgement e ence both both heat heat kernels kernels and solutions of heat equations. equations. As As an example, an example, 4 properties directly influence both heat kernels and solutions of heat equations. As an example, quations. Yet these and ble ible null null set, , ,and and .. ranslated lated asanset, follows. as follows. The author gratefully thanks Professor Sompong 5 Tasena’s result can be translated as follows. s. As example, cHarnack Harnackinequality, inequality, and and is isaanonincreasing nonincreasing remotely remotely 6 If (E,D(E)) satisfies parabolic Harnack inequality, Dhompongsa for several helpful comments and be abestrongly a strongly local regular regular Dirichlet Dirichlet form on onbe withwith associated associated 7 local Theorem 4. Let form a strongly local regular Dirichlet form on with associated Dirichlet Dirichlet space space satisfies satisfies parabolic parabolic Harnack Harnack and null a is a nonincreasing remotely constant function,. .suggestion. saaaconstant isconstant -accessible a -accessible null set, set, , and , and 8 such energy measure is a -accessible null set, , and . with associated such that that for for any any , and hand h then the weighted Dirichlet space (E ,D(E )) satisfiesremotely es isfies parabolic parabolic Harnack Harnack inequality, inequality, and and is a is nonincreasing a nonincreasing remotely 9 If satisfies parabolic Harnack inequality, and is a nonincreasing remotely . parabolic Harnack inequality if and only if there the weighted weighted Dirichlet Dirichlet space spacefunction, then satisfies satisfies parabolic parabolic Harnack Harnack 10 constant the weighted Dirichlet space satisfies parabolic Harnack References ncreasing remotely exists a constant such that for any and here if there exists exists a constant a constant such such that that for for any any and and 11 inequality if and only if there exists a constant such that for for anythe fundamental and parabolic Harnack Conclusion Conclusion Aronson, D.G. (1967). Bounds . .and 12 oduces roduces heat heat equations equations on on general general metric metric spaces spaces.using using solution of a parabolic equation. Bulletin of the generalization eneralization of of an an approach approach using using Sobolev Sobolev spaces spaces and and 13 American Mathematical Society, 73, 890-896. Conclusion Conclusion author uthor also alsodiscusses discusses when whenheat heatkernels kernelswill willhave haveaanice nice 14 Conclusion Conclusion Aronson, (1968). Non-negative solutionsspaces of linearusing the author author introduces introduces heatheat equations equations on aaon general general metric metric spaces spaces using usingD.G. kernel. ernel. This This isis15 ofofcourse course by byInno nothis means means complete complete list list ofof article, the author introduces heat equations on general metric a using In this article, the author introduces heat equations h. Thishowever, This is aisgeneralization generalization of an of approach an approach. approach using using Sobolev Sobolev spaces spaces andand parabolic equations.using Annali della spaces Scuola and will, will, however, provide provide an an introduction introduction toto this this field. field. 16 Dirichlet form This is a generalization of an approach Sobolev etric spaces unctions. ions.Dirichlet The The author author also discusses discusses when when heat heat kernels kernels will will have have a nice a nice bed rbed Dirichlet forms. Here Here the the author author introduces introduces weighted weighted onforms. general metric spaces using Dirichlet form approach. 17 weakly differentiable functions. The author also discusses when diheat will have obolev spaces andalso Normale Superiore Pisa,kernels 22(3), 607-694, 1968.a nice ssical classical heat heat kernel. kernel. This This is of is course of course by no by means no means a complete a complete list list of of fls Nash Nash (Nash, (Nash, 1958), 1958), Moser Moser (Moser, (Moser, 1961, 1961, 1964, 1964, 1967), 1967), 18 behavior similar to the classical heat kernel. This is of course by no means a complete list of will haveThis a nice is a generalization of an approach using Sobolev Aronson, D.G. (1971). Addendum: “non-negative solutions hopes es that that this this will, will, however, however, provide provide an introduction an introduction to this to this field. field. 971) 71) can canalso alsobe be considered results results ininauthor this thiscategory. category. Over Over 19considered references. The hopes that this will, however, provide an introduction to this field. a complete list ofand spaces weakly differentiable functions. The author weighted of linear parabolic Scuola. Norm. ays loways to perturbed to perturbed Dirichlet Dirichlet forms. forms. Here Here the the author author introduces introduces weighted been been made, made, many many assumptions assumptions have have been been weaken. weaken. Yet, Yet, 20 There are several ways to perturbed Dirichlet forms. Here equations” the author(Ann. introduces weighted this field. also discusses when heat kernels will have a nice behavior Sup. Pisa (3)22(1968), 607-694). Annali della Scuola original inal work work of Nash of Nash (Nash, (Nash, 1958), 1958), Moser Moser (Moser, (Moser, 1961, 1961, 1964, 1964, 1967), 1967), For For examples, examples, no one one knows knows sufficient sufficient and necessary necessary 21no Dirichlet spaces. Theand original work of Nash (Nash, 1958), Moser (Moser, 1961, 1964, 1967), ntroduces weighted similar tocan the classical heat kernel. This isinto oftothis course by1971) no can 67, 1967, 1968, 1968, 1971) 1971) can also also be considered beDirichlet considered results results in this category. category. Over be be doubling, doubling, or or for weighted weighted Dirichlet spaces spaces satisfy satisfy 22for and Aronson (Aronson, 1967, 1968, also be considered in this category. Over Normale Superiore diresults Pisa, 25(3), 221-228, 1971. Over 1961, 1964, 1967), ments vements have have been been made, made, many many assumptions assumptions have have been been weaken. weaken. Yet, Yet, ngularity ngularity set setmeans isisOver k-accessible, k-accessible, no nomany one one knows knows the the necessary 23 the years, improvements have been Bouleau, made, many assumptions been Forms weaken. a complete list of references. Thenecessary author hopes his category. N. & Hirsch, F. (1991).have Dirichlet andYet, nnanswered. unanswered. For For examples, examples, no no one one knows knows sufficient sufficient and and necessary necessary functions functions for for which which the the weighted weighted Dirichlet Dirichlet spaces spaces satisfy satisfy 24 many questions remain unanswered. For examples, no one knows sufficient and necessary been weaken. Yet, that this will, however, provide an introduction to this Analysis on Wiener Space. Berlin; New York: Walter measures be to 25 doubling, be doubling, or for or right for weighted weighted Dirichlet Dirichlet spaces spaces satisfy to satisfyor for weighted Dirichlet spaces to satisfy ndient neasures whether whether k-accessibility k-accessibility is isthe the right condition condition totoassume assume conditions for weighted measures to beto doubling, and tonecessary field. de is Gruyter. ven when when the to the singularity singularity set is setk-accessible, isinequality. k-accessible, no one no when one knows knows the the necessary necessary hese hese require require further further investigation investigation into into the the subject subject so so the 26 Poincare Even the singularity set k-accessible, no one knows the necessary et spaces satisfy There are several ways to perturbed Dirichlet Chen, Z.Q. & Fukushima, M.(2012).Dirichlet Symmetric Markov fnows semain the of the weight weight functions functions for for which which the the weighted weighted Dirichlet Dirichlet spaces spaces satisfy satisfy emain fruitful fruitful in in years years to to come. come. 27 and sufficient conditions of the weight functions for which the weighted spaces satisfy the necessary nequality. uality.spaces Even Even whether k-accessibility is the is inequality. the right right condition condition assume to assume forms. Here the author Harnack introduces weighted Dirichlet 28 whether thek-accessibility parabolic Eventowhether k-accessibility is the right to assume chlet satisfy Processes, Time Change, and condition Boundary Theory. nknown. own. All All these these require require further further investigation investigation into into the the subject subject so the so the Acknowledgement Acknowledgement 29 on singularity sets is unknown. All these require further investigation into the subject so the ondition to assume spaces. The original work of Nash (Nash, 1958), Moser Princeton University Press. sProfessor this field field will will remain remain fruitful fruitful in years in years to come. to come. Professor Sompong Dhompongsa Dhompongsa for forAronson several several helpful helpful author strongly believes this(Aronson, field will remain fruitful in years to come. the subjectSompong so30the1961, (Moser, 1964, 1967), and 1967, Grigor’yan, A. (2009). Heat Kernel and Analysis on 31 1968, 1971) can also be considered results in this category. Manifolds, volume 47 of AMS.IP Studies in Acknowledgement Acknowledgement 32 Acknowledgement Over the years, many improvements have been made, ully tefully thanks thanks Professor Sompong Sompong Dhompongsa Dhompongsa for several several helpful helpful Advanced Mathematics. American References References 33Professor The author gratefullyforthanks Professor Sompong Dhompongsa for Mathematical several helpful on. fundamental fundamental solution solution of of a a parabolic parabolic equation. equation. Bulletin Bulletin of of 34 assumptions comments and many havesuggestion. been weaken. Yet, many Society. International Press. or several helpful y,y,73, 73,890–896. 890–896. 35 questions remain unanswered. For examples, no one Grigor’yan, A. & Saloff-Coste, L. (2005). Stability results References olutions lutions ofoflinear linear parabolic parabolic equations. equations. Annali Annalidella dellaScuola Scuola 36References References knows sufficient and necessary conditions for weighted of forofHarnack inequality. Annales de L’Institut Bulletin Fourier, of unds Bounds for for the the fundamental solution solution aofparabolic a parabolic equation. equation. Bulletin 607–694, 607–694, 1968. 1968. 37fundamental Aronson, D.G.of(1967). Bounds for the Bulletin fundamental solution of a parabolic equation. measures to be doubling, or for weighted Dirichlet spaces Grenoble, 55(3), 825-890. atical ematical Society, Society, 73,of 73, 890–896. 890–896. on-negative n-negative solutions solutions of oflinear linear parabolic parabolic equations”(Ann. equations”(Ann. 38 the American Mathematical Society, 73, 890–896. quation. Bulletin to satisfy Poincare inequality. Even when the singularity Non-negative -negative solutions solutions of linear of linear parabolic parabolic equations. equations. 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