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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 60 (5 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.
SubGaussian estimates of heat kernels on infinite graphs
 Duke Math. J
, 2000
"... We prove that a two sided subGaussian estimate of the heat kernel on an infinite weighted graph takes place if and only if the volume growth of the graph is uniformly polynomial and the Green kernel admits a uniform polynomial decay. ..."
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Cited by 54 (10 self)
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We prove that a two sided subGaussian estimate of the heat kernel on an infinite weighted graph takes place if and only if the volume growth of the graph is uniformly polynomial and the Green kernel admits a uniform polynomial decay.
SaloffCoste L., Stability results for Harnack inequalities
"... We develop new techniques for proving uniform elliptic and parabolic Harnack inequalities on weighted Riemannian manifolds. In particular, we prove the stability of the Harnack inequalities under certain nonuniform changes of the weight. We also prove necessary and sufficient conditions for the Har ..."
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Cited by 27 (2 self)
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We develop new techniques for proving uniform elliptic and parabolic Harnack inequalities on weighted Riemannian manifolds. In particular, we prove the stability of the Harnack inequalities under certain nonuniform changes of the weight. We also prove necessary and sufficient conditions for the Harnack inequalities to hold on complete noncompact manifolds having nonnegative Ricci curvature outside a compact set and a finite first Betti number or just having asymptotically nonnegative sectional curvature. Contents
Ondiagonal lower bounds for heat kernels and Markov chains
 Duke Math. J
, 1997
"... Let M be a Riemannian manifold, and ∆ be the LaplaceBeltrami operator on M. It is known that there exists a unique minimal positive fundamental solution to the associated heat equation, which is referred to as the heat kernel and denoted by pt(x, y) (x, y ∈ M, t> 0). ..."
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Cited by 24 (2 self)
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Let M be a Riemannian manifold, and ∆ be the LaplaceBeltrami operator on M. It is known that there exists a unique minimal positive fundamental solution to the associated heat equation, which is referred to as the heat kernel and denoted by pt(x, y) (x, y ∈ M, t> 0).
The Art of Random Walks
 Lecture Notes in Mathematics 1885
, 2006
"... 1.1 Basic definitions and preliminaries................ 8 ..."
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Cited by 21 (4 self)
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1.1 Basic definitions and preliminaries................ 8
for the fundamental solution of elliptic and parabolic equations in nondivergence
"... In memory of Eugene Fabes Abstract. It is shown that any elliptic or parabolic operator in nondivergence form with measurable coefficients has a global fundamental solution verifying certain pointwise bounds. ..."
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Cited by 9 (0 self)
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In memory of Eugene Fabes Abstract. It is shown that any elliptic or parabolic operator in nondivergence form with measurable coefficients has a global fundamental solution verifying certain pointwise bounds.
Contents
, 1999
"... 2 Construction of the heat kernel on manifolds 3 2.1 Laplace operator.................................... 3 2.2 Eigenvalues and eigenfunctions of the Laplace operator............... 4 ..."
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2 Construction of the heat kernel on manifolds 3 2.1 Laplace operator.................................... 3 2.2 Eigenvalues and eigenfunctions of the Laplace operator............... 4