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Harnack inequalities and subGaussian estimates for random walks
 Math. Annalen
, 2002
"... We show that a fiparabolic Harnack inequality for random walks on graphs is equivalent, on one hand, to so called fiGaussian estimates for the transition probability and, on the other hand, to the conjunction of the elliptic Harnack inequality, the doubling volume property, and the fact that the m ..."
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Cited by 30 (6 self)
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We show that a fiparabolic Harnack inequality for random walks on graphs is equivalent, on one hand, to so called fiGaussian estimates for the transition probability and, on the other hand, to the conjunction of the elliptic Harnack inequality, the doubling volume property, and the fact that the mean exit time in any ball of radius R is of the order R . The latter condition can be replaced by a certain estimate of a resistance of annuli.
On the relation between elliptic and parabolic Harnack inequalities
, 2001
"... We show that, if a certain Sobolev inequality holds, then a scaleinvariant elliptic Harnack inequality suces to imply its a priori stronger parabolic counterpart. Neither the relative Sobolev inequality nor the elliptic Harnack inequality alone suces to imply the parabolic Harnack inequality in que ..."
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Cited by 27 (3 self)
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We show that, if a certain Sobolev inequality holds, then a scaleinvariant elliptic Harnack inequality suces to imply its a priori stronger parabolic counterpart. Neither the relative Sobolev inequality nor the elliptic Harnack inequality alone suces to imply the parabolic Harnack inequality in question; both are necessary conditions. As an application, we show the equivalence between parabolic Harnack inequality for on M , (i.e., for @ t + ) and elliptic Harnack inequality for @ 2 t + on R M . 1
Dirichlet Heat Kernel in the Exterior of a Compact Set
"... this paper do not directly use the Ricci curvature assumption. In fact, we will show that Theorems 1.1, 1.2 hold true for any manifold which is quasiisometric (even roughlyisometric, under any reasonable bounded local geometry assumption) to a manifold with nonnegative Riccicurvature. In particu ..."
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Cited by 5 (1 self)
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this paper do not directly use the Ricci curvature assumption. In fact, we will show that Theorems 1.1, 1.2 hold true for any manifold which is quasiisometric (even roughlyisometric, under any reasonable bounded local geometry assumption) to a manifold with nonnegative Riccicurvature. In particular, the bounds of Examples 1.1, 1.2, hold true if the Laplace operator is replaced by a uniformly elliptic operator in divergence form. Thus, the bounds stated above are reasonably stable. The adequate hypothesis for our purpose is expressed in terms of a parabolic Harnack inequality or, equivalently, in terms of certain Poincar'e inequality and volume growth (see below Section 2.2). The present work originated from our desire to understand the behavior of the heat kernel on manifolds with more than one ends. Indeed, together with good estimates of certain hitting probabilities obtained in [14], the result presented here is one of the main building blocks in the proof of the sharp estimates for the heat kernel on manifolds with ends that have been announced in [11] and are proved in [12]. The following result complements Theorems 1.1, 1.2 in this direction. Given a Riemannian manifold with k ends, let U be a relatively compact open set in M with smooth boundary such that M n U has exactly k unbounded connected components E 1 ; : : : ; E k . Let K i = @U " E i , and consider E i as a manifold with boundary ffi E i := K i . Denote by p i the heat kernel on E i and by p\Omega i the Dirichlet heat kernel on\Omega i = E i n K i (in other words, p i satisfies the Neumann condition on K i , whereas p\Omega i satisfies the Dirichlet condition on K i ). Let also V i (x; t) be the volume function on E i . For each end E i , fix a point o i 2 K i and define the functions H i , D...
Sobolev inequalities in familiar and unfamiliar settings
 In S. Sobolev Centenial Volumes, (V. Maz’ja, Ed
"... Abstract The classical Sobolev inequalities play a key role in analysis in Euclidean spaces and in the study of solutions of partial differential equations. In fact, they are extremely flexible tools and are useful in many different settings. This paper gives a glimpse of assortments of such applica ..."
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Abstract The classical Sobolev inequalities play a key role in analysis in Euclidean spaces and in the study of solutions of partial differential equations. In fact, they are extremely flexible tools and are useful in many different settings. This paper gives a glimpse of assortments of such applications in a variety of contexts. 1
Asymptotic separation for independent trajectories of Markov processes
"... Markov processes 15 7 Examples 18 7.1 Diffusions on manifolds : : : : : : : : : : : : : : : : : : : : : : : : : : 19 7.2 The ffprocess : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 21 7.3 Random walk : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 23 8 Proofs 24 8.1 ..."
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Cited by 1 (0 self)
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Markov processes 15 7 Examples 18 7.1 Diffusions on manifolds : : : : : : : : : : : : : : : : : : : : : : : : : : 19 7.2 The ffprocess : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 21 7.3 Random walk : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 23 8 Proofs 24 8.1 Intersections of trajectories with covering balls : : : : : : : : : : : : : 24 8.2 Hitting probability and Green kernel : : : : : : : : : : : : : : : : : : 27 8.3 Asymptotic separation in terms of the Green kernel : : : : : : : : : : 28 8.4 Asymptotic separation for two trajectories : : : : : : : : : : : : : : : 31 Supported by the EPSRC Research Fellowship B/94/AF/1782 y Partially supported by the EPSRC Visiting Fellowship GR/M61573 1 8.5 Asymptotic separation for n trajectories : : : : : : : : : : : : : : : : 33 1
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 MSJSI 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
RIESZ TRANSFORMS ON CONNECTED SUMS
, 2006
"... Let (M, g) be a complete Riemannian manifold with infinite volume, we denote by ∆ = ∆ g its Laplace operator, it has an unique selfadjoint extension on L 2 (M, dvolg) which is also denoted by ∆. The Green formula and the spectral theorem show that for any ϕ ∈ C ∞ 0 (M): ..."
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Let (M, g) be a complete Riemannian manifold with infinite volume, we denote by ∆ = ∆ g its Laplace operator, it has an unique selfadjoint extension on L 2 (M, dvolg) which is also denoted by ∆. The Green formula and the spectral theorem show that for any ϕ ∈ C ∞ 0 (M):
(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
Probabilistic Approach to Geometry Heat kernel estimates, III
"... lim t→0 (−t log Pµ(X0 ∈ A & Xt ∈ B)) = d(A, B)2 4 title Manifolds with finitely many ends The heat kernel on Manifolds with ends Gluing techniques The general transient case Manifolds with ends We will consider manifolds with ends: M = M1#M2 #... #Mk where the ends Mi, 1 ≤ i ≤ k are of Harnack type. ..."
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lim t→0 (−t log Pµ(X0 ∈ A & Xt ∈ B)) = d(A, B)2 4 title Manifolds with finitely many ends The heat kernel on Manifolds with ends Gluing techniques The general transient case Manifolds with ends We will consider manifolds with ends: M = M1#M2 #... #Mk where the ends Mi, 1 ≤ i ≤ k are of Harnack type. M = K ∪ E1 ∪ · · · ∪ Ek (disjoint union) with K compact with smooth boundary and Ei isometric to an open set in Mi (we can allow Ei = Mi). Curvature conditions that yield such manifolds: (c1) Asymptotically nonnegative sectional curvature, (c2) Nonnegative Ricci curvature outside a compact set with ends satisfying (RCA) title Manifolds with finitely many ends The heat kernel on Manifolds with ends Gluing techniques The general transient case Euclidean domainstitle Manifolds with finitely many ends The heat kernel on Manifolds with ends Gluing techniques The general transient case The heat kernel on manifolds with ends Consider M = M1 # · · · #Mk and assume that each Mk is of Harnack type, transient. Then the heat kernel is bounded above and below by expressions of the type 1