## The heat kernel and its estimates

### BibTeX

@MISC{Saloff-coste_theheat,

author = {Laurent Saloff-coste},

title = {The heat kernel and its estimates},

year = {}

}

### OpenURL

### Abstract

After a short survey of some of the reasons that make the heat kernel an important object of study, we review a number of basic heat kernel estimates. We then describe recent results concerning (a) the heat kernel on certain manifolds with ends, and (b) the heat kernel with Neumann or Dirichlet boundary condition in Euclidean domains. This text is a revised version of the four lectures given by the author at the First MSJ-SI in Kyoto during the summer of 2008. The structure of the lectures has been mostly preserved although some material has been added, deleted, or shifted around. The goal is to present an

### Citations

412 | Theory of Function Spaces - Triebel - 1983 |

210 | Dirichlet forms and symmetric Markov processes - Fukushima, Ōshima, et al. - 1994 |

151 |
On the parabolic kernel of the Schrödinger operator
- Li, Yau
- 1986
(Show Context)
Citation Context ... (9) c(ɛ) V (x, √ t) d(x,y)2 e− 4(1+ɛ)t ≤ p(t, x, y) ≤ C(ɛ) V (x, √ d(x,y)2 e− 4(1−ɛ)t , t) as well as the companion gradient estimate (10) |∇yp(t, x, y)| ≤ C(ɛ) d(x,y)2 − √ √ e 4(1−ɛ)t tV (x, t) See =-=[42]-=-. 3.2. The characterization of (PHI) In contrast to the fact that we do not have a precise description of those complete weighted Riemannian manifolds that satisfy the elliptic Harnack inequality, the... |

142 |
Heat kernels and spectral theory”, Cambridge Tracts
- Davies
- 1990
(Show Context)
Citation Context ...llustrated by the following result due to A. Grigor’yan which is typical of an important body of work developed by E.B. Davies, N. Varopoulos, A. Grigor’yan and others concerning Gaussian bounds. See =-=[13, 16, 21, 26, 27, 50, 62]-=-. Given a continuous decreasing function v, we consider the condition (∗) ∃ 0 < a < 1 < A < ∞, u(As) > au(s), u = v ′ /v. This condition means that the logarithmic derivative u of v decays at most as ... |

114 |
Continuity of solutions of parabolic and elliptic equations
- Nash
- 1958
(Show Context)
Citation Context ... nature of the equation. As in the elliptic case, the parabolic Harnack inequality (8) implies the Hölder continuity of the corresponding local solution (this continuity was first obtained by Nash in =-=[45]-=-). Again, in the context of weighted Riemannian manifolds, we may consider (8) as a property, call it (PHI), that may or may not be satisfied. For complete Riemannian manifolds (M, g) of dimension n w... |

89 |
Differential equations on Riemannian manifolds and their geometric applications
- Cheng, Yau
- 1975
(Show Context)
Citation Context ...ality as a property that may or may not be satisfied. It is an open question to characterize (in useful terms) those weighted manifolds that satisfy this property. However, around 1975, Cheng and Yau =-=[16]-=- proved that on any complete Riemannian manifold (M, g) with Ricci curvature bounded below by Ric ≥ −Kg, for some K ≥ 0, any positive harmonic function in a ball B = B(x, r) satisfies |∇ log u| ≤ C(n)... |

84 |
On Harnack’s theorem for elliptic differential equations
- Moser
- 1961
(Show Context)
Citation Context ... (elliptic) Harnack inequality. Its best known consequence is the fact that global positive harmonic functions in R n must be constant.hk.tex : 2008/12/5 (10:17) page: 10 10 L. Saloff-Coste J. Moser =-=[43]-=- observed that (7) also holds for uniformly elliptic operators (with measurable coefficients) in divergence form in R n and that this implies the crucial Hölder continuity property of the local soluti... |

65 |
A Harnack inequality for parabolic differential equations
- Moser
- 1964
(Show Context)
Citation Context ..., this gradient Harnack estimate immediately implies the validity of (7). 3.1.2. The parabolic Harnack inequality (PHI) The parabolic version of (7) is attributed by J. Moser to Hadamard and Pini. In =-=[44]-=-, J. Moser proved the parabolic Harnack inequality for uniformly elliptic operators in divergence form in Rn . This inequality states that a positive solution u of the heat equation in a cylinder of t... |

59 |
Upper bounds for symmetric Markov transition functions
- Carlen, Kusuoka, et al.
- 1987
(Show Context)
Citation Context ...t, x, y)} ≤ Ct x,y −n/2 , t > 0, and the Sobolev inequality ∫ ≤ S (6) ‖f‖ 2 2n/(n−2) M |∇f| 2 dµ, f ∈ C ∞ c (M). Other developments in this direction are recorded in the following theorem. See, e.g., =-=[13, 26, 41, 47, 50, 52, 62]-=-.hk.tex : 2008/12/5 (10:17) page: 7 Heat kernel 7 Theorem 2.1. For any fixed n > 0, the bound (5) is equivalent to each the following properties: • The Nash inequality: ∀ f ∈ C ∞ c (M), ‖f‖ 2(1+2/n) ... |

57 |
The heat equation on non compact Riemannian manifold
- Grigory’an
- 1992
(Show Context)
Citation Context ... complete weighted Riemannian manifolds satisfying the parabolic version (PHI). This theorem also applies to weighted manifolds with boundary as long as the Neumann condition is assumed. Theorem 3.1 (=-=[25, 48]-=-). Let (M, g) be a weighted complete Riemannian manifold. The following three properties are equivalent: • The parabolic Harnack inequality (PHI).hk.tex : 2008/12/5 (10:17) page: 12 12 L. Saloff-Cost... |

56 |
Analytic and geometric background of recurrence and non-explosion of the Brownian motion on Riemannian manifolds
- Grigor’yan
- 1999
(Show Context)
Citation Context ...llustrated by the following result due to A. Grigor’yan which is typical of an important body of work developed by E.B. Davies, N. Varopoulos, A. Grigor’yan and others concerning Gaussian bounds. See =-=[13, 16, 21, 26, 27, 50, 62]-=-. Given a continuous decreasing function v, we consider the condition (∗) ∃ 0 < a < 1 < A < ∞, u(As) > au(s), u = v ′ /v. This condition means that the logarithmic derivative u of v decays at most as ... |

52 |
A note on the isoperimetric constant
- Buser
- 1982
(Show Context)
Citation Context ... Riemannian manifolds with non-negative Ricci curvature. In this case, the doubling property follows from the celebrated Bishop-Gromov volume comparison. The Poincaré inequality follows from the work =-=[11]-=- of P. Buser. See also [50, Sect. 5.6.3]. The parabolic Harnack inequality and the twosided heat kernel bound were first obtained by Li and Yau in [42]. • Convex domains in Euclidean space. The doubli... |

49 | Non-negative solutions of linear parabolic equations - Aronson - 1968 |

49 |
Aspects of Sobolev-type inequalities
- Saloff-Coste
(Show Context)
Citation Context ...t, x, y)} ≤ Ct x,y −n/2 , t > 0, and the Sobolev inequality ∫ ≤ S (6) ‖f‖ 2 2n/(n−2) M |∇f| 2 dµ, f ∈ C ∞ c (M). Other developments in this direction are recorded in the following theorem. See, e.g., =-=[13, 26, 41, 47, 50, 52, 62]-=-.hk.tex : 2008/12/5 (10:17) page: 7 Heat kernel 7 Theorem 2.1. For any fixed n > 0, the bound (5) is equivalent to each the following properties: • The Nash inequality: ∀ f ∈ C ∞ c (M), ‖f‖ 2(1+2/n) ... |

44 | Bounds for the fundamental solution of a parabolic equation - Aronson - 1967 |

44 |
A note on Poincaré, Sobolev and Harnack inequalities
- Saloff-Coste
- 1992
(Show Context)
Citation Context ... complete weighted Riemannian manifolds satisfying the parabolic version (PHI). This theorem also applies to weighted manifolds with boundary as long as the Neumann condition is assumed. Theorem 3.1 (=-=[25, 48]-=-). Let (M, g) be a weighted complete Riemannian manifold. The following three properties are equivalent: • The parabolic Harnack inequality (PHI).hk.tex : 2008/12/5 (10:17) page: 12 12 L. Saloff-Cost... |

42 |
Saloff-Coste L., Isopérimétrie pour les groupes et les variétés
- Coulhon
- 1993
(Show Context)
Citation Context ...r ≥ 1. Then p(t, x, x) ≥ c , t ≥ 1. (t log t) N/2 These bounds are (essentially) sharp in the sense that there are examples where they describe (essentially) the true behavior of the heat kernel. See =-=[9, 17, 18]-=- for details. The following result complement the previous theorem by giving an important collection of examples where the long time behavior of the heat kernel is tightly connected to the volume grow... |

42 |
Heat kernel upper bounds on a complete non-compact manifold
- Grigor’yan
- 1994
(Show Context)
Citation Context ...t, x, y)} ≤ Ct x,y −n/2 , t > 0, and the Sobolev inequality ∫ ≤ S (6) ‖f‖ 2 2n/(n−2) M |∇f| 2 dµ, f ∈ C ∞ c (M). Other developments in this direction are recorded in the following theorem. See, e.g., =-=[13, 26, 41, 47, 50, 52, 62]-=-.hk.tex : 2008/12/5 (10:17) page: 7 Heat kernel 7 Theorem 2.1. For any fixed n > 0, the bound (5) is equivalent to each the following properties: • The Nash inequality: ∀ f ∈ C ∞ c (M), ‖f‖ 2(1+2/n) ... |

34 |
Rough isometries and combinatorial approximations of geometries of noncompact riemannian manifolds
- Kanai
- 1985
(Show Context)
Citation Context ...pirit. • Any weighted complete Riemannian manifold with bounded geometry (see Section 2.4) which is (volume) quasi-isometric to a complete weighted manifold satisfying (PHI) also satisfies (PHI). See =-=[40, 19]-=-. 3.4. Some consequences In this subsection, I describe two consequences of the parabolic Harnack inequality (PHI) that will play a role in Lecture III. They concern the Dirichlet heat kernel in the c... |

33 | Riesz transform on manifolds and heat kernel regularity - Auscher, Coulhon, et al. |

33 |
Hardy-Littlewood Theory for Semigroups
- Varopoulos
- 1985
(Show Context)
Citation Context ...be the associated heat kernel, i.e., the kernel associated to the heat semigroup e t∆ on L 2 (M, µ). 2.3.1. Varopoulos’ theorem Although there are various antecedents in the literature, N. Varopoulos =-=[60]-=- was the first to identify the equivalence between heat kernel bounds and Sobolev inequality in a large enough context. In particular, Varopoulos proved, for any fixed n > 2, the equivalence between t... |

32 |
On the behavior of the fundamental solution of the heat equation with variable coefficients
- VARADHAN
- 1967
(Show Context)
Citation Context ...ons between the fields of Analysis, Probability and Geometry and this will be illustrated here again. The simplest statement embodying these interactions is perhaps Varadhan’s large deviation formula =-=[59]-=- lim t→0 −4t log p(t, x, y) = d(x, y)2 . This formula relates the heat kernel p(t, x, y) to the Riemannian distance function d(x, y) on a complete Riemannian manifold. A remarkable generalization was ... |

30 | Non-Gaussian aspects of heat kernel behaviour - Davies - 1997 |

29 |
G.,Inégalités isopérimétriques de Faber-Krahn et conséequences, Actes de la table ronde de géométrie différentielle (Luminy
- Carron
- 1992
(Show Context)
Citation Context ... negative L2-eigenvalues of A. Note that the lowest Dirichlet eigenvalue in a open set Ω is defined by the variational formula {∫ 2 |∇φ| dµ λ(Ω) = inf ∫ : φ ∈ C |φ| 2dµ ∞ } c (Ω), φ ̸= 0 . See, e.g., =-=[14, 26]-=-. For a topological application of the RCL inequality, see [15]. 2.3.2. The general Gaussian upper bound One of the most important discovery in the area of heat kernel estimates is the “universality” ... |

28 | Heat kernels on weighted manifolds and applications - Grigor’yan - 2006 |

26 |
Variétés Riemanniennes isométriques à l’infini
- Coulhon, Saloff-Coste
- 1995
(Show Context)
Citation Context ...roup Γ. The group Γ is finitely generated and, choosing a finite symmetric generating set, we can consider its volume growth. If Γ has polynomial volume growth then (M, g) satisfies (PHI). See, e.g., =-=[19, 49, 51]-=-. • Assume that M, N are two complete Riemannian manifolds and G is a group of isometries of M such that M/G = N. Then, if M satisfies (PHI) so does N. • Consider the Euclidean space Rn , n ≥ 2, with ... |

26 | On the relation between elliptic and parabolic Harnack inequalities
- Hebisch, Saloff-Coste
(Show Context)
Citation Context ...on of the class of manifold that satisfy (PHI) is as follows: A complete weighted manifold M satisfies (PHI) if and only if the Riemannian product R × M satisfies the elliptic Harnack inequality. See =-=[37]-=- for details and further results. B kBhk.tex : 2008/12/5 (10:17) page: 13 Heat kernel 13 3.3. Examples of weighted manifolds satisfying (PHI) Here is a list of examples with additional comments in ea... |

25 | Local behavior of solutions of quasi-linear parabolic equations. Archive for Rational Mechanics and Analysis - Aronson, Serrin - 1967 |

25 |
Ultracontractivity and Nash type inequalities
- Coulhon
- 1996
(Show Context)
Citation Context ...llustrated by the following result due to A. Grigor’yan which is typical of an important body of work developed by E.B. Davies, N. Varopoulos, A. Grigor’yan and others concerning Gaussian bounds. See =-=[13, 16, 21, 26, 27, 50, 62]-=-. Given a continuous decreasing function v, we consider the condition (∗) ∃ 0 < a < 1 < A < ∞, u(As) > au(s), u = v ′ /v. This condition means that the logarithmic derivative u of v decays at most as ... |

24 | Manifolds and graphs with slow heat kernel decay - Barlow, Coulhon, et al. |

23 |
Analysis on local Dirichlet spaces. I: Recurrence, conservativeness and L p -Liouville properties
- Sturm
(Show Context)
Citation Context ...of Theorem 3.1 to the abstract setting of regular strictly local Dirichlet spaces. This extension was obtained byhk.tex : 2008/12/5 (10:17) page: 27 Heat kernel 27 K. Th. Sturm in a series of papers =-=[53, 54, 55]-=-. See also the related work of Biroli and Mosco [10]. Without entering into the details, a Harnack-type Dirichlet space is a regular strictly local Dirichlet space (M, λ, E, F) with the following addi... |

22 |
Analysis on local Dirichlet spaces. III. The parabolic Harnack inequality
- Sturm
- 1996
(Show Context)
Citation Context ...of Theorem 3.1 to the abstract setting of regular strictly local Dirichlet spaces. This extension was obtained byhk.tex : 2008/12/5 (10:17) page: 27 Heat kernel 27 K. Th. Sturm in a series of papers =-=[53, 54, 55]-=-. See also the related work of Biroli and Mosco [10]. Without entering into the details, a Harnack-type Dirichlet space is a regular strictly local Dirichlet space (M, λ, E, F) with the following addi... |

20 |
A Saint-Venant type principle for Dirichlet forms on discontinuous
- Biroli, Mosco
- 1995
(Show Context)
Citation Context ...ocal Dirichlet spaces. This extension was obtained byhk.tex : 2008/12/5 (10:17) page: 27 Heat kernel 27 K. Th. Sturm in a series of papers [53, 54, 55]. See also the related work of Biroli and Mosco =-=[10]-=-. Without entering into the details, a Harnack-type Dirichlet space is a regular strictly local Dirichlet space (M, λ, E, F) with the following additional properties: • (a) M is a locally compact conn... |

18 |
On-diagonal lower bounds for heat kernels and Markov chains, Duke
- Coulhon, Grigor’yan
- 1997
(Show Context)
Citation Context ...r ≥ 1. Then p(t, x, x) ≥ c , t ≥ 1. (t log t) N/2 These bounds are (essentially) sharp in the sense that there are examples where they describe (essentially) the true behavior of the heat kernel. See =-=[9, 17, 18]-=- for details. The following result complement the previous theorem by giving an important collection of examples where the long time behavior of the heat kernel is tightly connected to the volume grow... |

17 | Theorie du potentiel sur des groupes et des varietes - VAROPOULOS - 1986 |

16 |
The Rozenblum-Lieb-Cwikel inequality for Markov generators
- Levin, Solomyak
- 1997
(Show Context)
Citation Context |

16 |
Analysis on local Dirichlet spaces II. Upper Gaussian estimates for the fundamental solution of parabolic equations
- Sturm
- 1995
(Show Context)
Citation Context ...of Theorem 3.1 to the abstract setting of regular strictly local Dirichlet spaces. This extension was obtained byhk.tex : 2008/12/5 (10:17) page: 27 Heat kernel 27 K. Th. Sturm in a series of papers =-=[53, 54, 55]-=-. See also the related work of Biroli and Mosco [10]. Without entering into the details, a Harnack-type Dirichlet space is a regular strictly local Dirichlet space (M, λ, E, F) with the following addi... |

15 |
Plancherel-type estimates and sharp spectral multipliers
- Duong, Ouhabaz, et al.
(Show Context)
Citation Context ... − ∆) −1 , if it exists) Gα(x, y) = ∫ ∞ 0 e −αt p(t, x, y)dt, α ≥ 0. A more sophisticated application concerns the action of the imaginary power of the Laplacian (−∆) iβ on the L p spaces. See, e.g., =-=[20, 23]-=-. In a similar spirit, heat kernel bounds play a crucial role in the study of the boundedness of Riesz transforms on L p spaces on manifolds. See [7] and Hofmann article in [39]. 2.2.3. Applications t... |

13 | Two-sided estimates for fundamental solutions of second-order parabolic equations, and some applications Uspekhi - Eidel’man, Porper - 1984 |

12 |
Analysis and geometry on groups, volume 100 of Cambridge Tracts in Mathematics
- Varopoulos, Coulhon
- 1992
(Show Context)
Citation Context |

11 |
Spaces of harmonic functions
- Sung, Tam, et al.
(Show Context)
Citation Context ... M = M1# . . . #Mk is transient with each Mi satisfying (PHI). We assume further that each Mi satisfies (RCA) (see Defnition 3.3). Then there exists a positive harmonic function h on M such that (see =-=[30, 57]-=-) ( ∫ 2 |x| ds h(x) ≃ 1 + Vix (√ ) . s) 1 Theorem 4.2 ([32, 33]). The ends ˜ Mi of the weighted manifold + are transient and satisfy (PHI). ˜M = (M, h 2 dµ) = ˜ M1# . . . # ˜ Mk The classical Doob tra... |

10 |
and LSC: Heat kernel on connected sums of Riemannian manifolds
- Grigor’yan
- 1999
(Show Context)
Citation Context ...ates in the recurrent case are very interesting but we refer the reader to [31, Sec. 4.4]. §4. Lecture III: The heat kernel on manifold with ends The aim of this Lecture is to describe the results of =-=[29, 33]-=- which provide two-sided estimates that are fundamentally different from (11). This is joint work with A. Grigor’yan. We investigate the heat kernel on complete weighted manifolds of the form M = M1# ... |

10 |
Parabolic Harnack inequality for divergence-form second-order differential operators
- Saloff-Coste
- 1995
(Show Context)
Citation Context ... (M), |f − fB| 2 dµ ≤ P r 2 ∫ |∇f| 2 dµ, B where B = B(x, r), kB = B(x, kr), x ∈ M, r > 0 and fB is the mean of f over B. It is in this weaker form that the Poincaré inequality is often obtained. See =-=[25, 48, 49]-=- and also [50, Chapter 5]. The doubling property implies that for all x, y ∈ M and 0 < r < R < ∞ ( ) α V (y, R) d(x, y) + R ≤ D1 V (x, s) r and, if M is assumed to be non-compact, V (x, R) V (x, s) ≥ ... |

9 |
Small-time Gaussian behavior of symmetric diffusion semigroups
- Hino, Ramírez
(Show Context)
Citation Context ...x, y)2 . This formula relates the heat kernel p(t, x, y) to the Riemannian distance function d(x, y) on a complete Riemannian manifold. A remarkable generalization was given by Hino and Ramirez (see, =-=[3, 38]-=-). Namely, in the context of Dirichlet spaces, lim t→0 −4t log P(X0 ∈ A; Xt ∈ B) = d(A, B) 2 . Here Xt denotes the Markov process associated with the underlying local regular Dirichlet form and d is t... |

9 |
Analysis on Riemannian co-compact covers
- SALOFF-COSTE
- 2004
(Show Context)
Citation Context ...r) ≃ rD , r ∈ (1, ∞), and the heat kernel satisfies p(t, , x, y) ≃ t−D/2 , t ∈ (1, ∞). • Or log V (r) ≃ r, r ∈ (1, ∞), and the heat kernel satisfies − log(p(t, , x, y)) ≃ t1/3 , t ∈ (1, ∞)). See also =-=[51, 52]-=- for a further review of results in this direction and references. The second case, i.e., for amenable unimodular connected Lie groups of exponential volume growth, the estimate take the form exp ( −c... |

8 |
Saloff-Coste L., Stability results for Harnack inequalities
- Grigor’yan
(Show Context)
Citation Context ...ace Rn , n ≥ 2, with weight (1 + |x| 2 ) α/2 , α ∈ (−∞, ∞). This is a complete weighted manifold. It satisfies (PHI) if and only if α > −n. It satisfies the elliptic Harnack inequality for all α. See =-=[32]-=- for this and other examples in this spirit. • Any weighted complete Riemannian manifold with bounded geometry (see Section 2.4) which is (volume) quasi-isometric to a complete weighted manifold satis... |

7 | Gaussian heat kernel upper bounds via the Phragmén-Lindelöf theorem
- Coulhon, Sikora
(Show Context)
Citation Context ... − ∆) −1 , if it exists) Gα(x, y) = ∫ ∞ 0 e −αt p(t, x, y)dt, α ≥ 0. A more sophisticated application concerns the action of the imaginary power of the Laplacian (−∆) iβ on the L p spaces. See, e.g., =-=[20, 23]-=-. In a similar spirit, heat kernel bounds play a crucial role in the study of the boundedness of Riesz transforms on L p spaces on manifolds. See [7] and Hofmann article in [39]. 2.2.3. Applications t... |

7 |
The Cwikel-Lieb-Rozenblum estimates for generators of positive semigroups and semigroups dominated by positive semigroups, (Russian) Algebra i Analiz 9
- Rozenblum, Solomyak
- 1997
(Show Context)
Citation Context |

6 |
Semi-groups of operators and approximation. Die Grundlehren der mathematischen Wissenschaften, Band 145
- Butzer, Berens
- 1967
(Show Context)
Citation Context ...: 2008/12/5 (10:17) page: 5 Heat kernel 5 Similarly, one of the equivalent definition of the Hardy space H1 (whose dual is BMO), is that H1 = {f ∈ L 1 (R n ) : sup |Htf| ∈ L t>0 1 (R n )}. See, e.g., =-=[12]-=- and [58, Chapter I]. The whole scales of Besov and Lizorkin spaces can be treated in similar ways. One of the point we want to make here is that these versions of the definitions of classical functio... |

5 | Boundary Harnack principle and Martin boundary for a uniform domain - Aikawa |