### The heat kernel and its estimates

"... 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 boun ..."

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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

### ON THE REGULARITY OF SAMPLE PATHS OF SUB-ELLIPTIC DIFFUSIONS ON MANIFOLDS

, 2004

"... Using heat kernel Gaussian estimates and related properties, we study the intrinsic regularity of the sample paths of the Hunt process associated to a strictly local regular Dirichlet form. We describe how the results specialize to Riemannian Brownian motion and to sub-elliptic symmetric diffusions. ..."

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Using heat kernel Gaussian estimates and related properties, we study the intrinsic regularity of the sample paths of the Hunt process associated to a strictly local regular Dirichlet form. We describe how the results specialize to Riemannian Brownian motion and to sub-elliptic symmetric diffusions. 1.

### unknown title

, 2010

"... www.elsevier.com/locate/jfa Comparison inequalities for heat semigroups and heat kernels on metric measure spaces ✩ Alexander Grigor’yan a, Jiaxin Hu b, ∗ , Ka-Sing Lau c ..."

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www.elsevier.com/locate/jfa Comparison inequalities for heat semigroups and heat kernels on metric measure spaces ✩ Alexander Grigor’yan a, Jiaxin Hu b, ∗ , Ka-Sing Lau c

### Obtaining Upper . . . From Lower Bounds

, 2007

"... We show that a near-diagonal lower bound of the heat kernel of a Dirichlet form on a metric measure space with a regular measure implies an on-diagonal upper bound. If in addition the Dirichlet form is local and regular, then we obtain a full off-diagonal upper bound of the heat kernel provided the ..."

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We show that a near-diagonal lower bound of the heat kernel of a Dirichlet form on a metric measure space with a regular measure implies an on-diagonal upper bound. If in addition the Dirichlet form is local and regular, then we obtain a full off-diagonal upper bound of the heat kernel provided the Dirichlet heat kernel on any ball satisfies a near-diagonal lower estimate. This reveals a new phenomenon in the relationship between the lower and upper bounds of the heat kernel.

### GREEN KERNEL ESTIMATES AND THE FULL MARTIN BOUNDARY FOR RANDOM WALKS ON LAMPLIGHTER GROUPS AND DIESTEL-LEADER GRAPHS

, 2004

"... Abstract. We determine the precise asymptotic behaviour (in space) of the Green kernel of simple random walk with drift on the Diestel-Leader graph DL(q, r), where q, r ≥ 2. The latter is the horocyclic product of two homogeneous trees with respective degrees q + 1 and r + 1. When q = r, it is the C ..."

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Abstract. We determine the precise asymptotic behaviour (in space) of the Green kernel of simple random walk with drift on the Diestel-Leader graph DL(q, r), where q, r ≥ 2. The latter is the horocyclic product of two homogeneous trees with respective degrees q + 1 and r + 1. When q = r, it is the Cayley graph of the wreath product (lamplighter group) Zq ≀ Z with respect to a natural set of generators. We describe the full Martin compactification of these random walks on DL-graphs and, in particular, lamplighter groups. This completes and provides a better approach to previous results of Woess, who has determined all minimal positive harmonic functions. 1.

### Localized BMO and BLO Spaces on RD-Spaces and Applications to Schrödinger Operators

, 903

"... Abstract. An RD-space X is a space of homogeneous type in the sense of Coifman and Weiss with the additional property that a reverse doubling condition holds in X. Let ρ be an admissible function on RD-space X. The authors first introduce the localized spaces BMOρ(X) and BLOρ(X) and establish their ..."

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Abstract. An RD-space X is a space of homogeneous type in the sense of Coifman and Weiss with the additional property that a reverse doubling condition holds in X. Let ρ be an admissible function on RD-space X. The authors first introduce the localized spaces BMOρ(X) and BLOρ(X) and establish their basic properties, including the John-Nirenberg inequality for BMOρ(X), several equivalent characterizations for BLOρ(X), and some relations between these spaces. Then the authors obtain the boundedness on these localized spaces of several operators including the natural maximal operator, the Hardy-Littlewood maximal operator, the radial maximal functions and their localized versions associated to ρ, and the Littlewood-Paley g-function associated to ρ, where the Littlewood-Paley g-function and some of the radial maximal functions are defined via kernels which are modeled on the semigroup generated by the Schrödinger operator. These results apply in a wide range of settings, for instance, to the Schrödinger operator or the degenerate Schrödinger operator on R d, or the sub-Laplace Schrödinger operator on Heisenberg groups or connected and simply connected nilpotent Lie groups. 1

### unknown title

"... www.imstat.org/aihp Random walk on graphs with regular resistance and volume growth ..."

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www.imstat.org/aihp Random walk on graphs with regular resistance and volume growth

### (a) The conjunction of • The doubling property: V (x, 2r) ≤ DV (x, r), for all x, r. • The Poincaré inequality: For all B = B(x, r),

"... Heat kernel estimates, II lim t→0 (−t log Pµ(X0 ∈ A & Xt ∈ B)) = d(A, B)2 4 title Scale and location homogeneity Manifolds with finitely many ends The heat kernel on Manifolds with ends ..."

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Heat kernel estimates, II lim t→0 (−t log Pµ(X0 ∈ A & Xt ∈ B)) = d(A, B)2 4 title Scale and location homogeneity Manifolds with finitely many ends The heat kernel on Manifolds with ends