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27
Heat kernel estimates for jump processes of mixed types on metric measure spaces
 FIELDS
"... In this paper, we investigate symmetric jumptype processes on a class of metric measure spaces with jumping intensities comparable to radially symmetric functions on the spaces. The class of metric measure spaces includes the Alfors dregular sets, which is a class of fractal sets that contains ge ..."
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Cited by 50 (30 self)
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In this paper, we investigate symmetric jumptype processes on a class of metric measure spaces with jumping intensities comparable to radially symmetric functions on the spaces. The class of metric measure spaces includes the Alfors dregular sets, which is a class of fractal sets that contains geometrically selfsimilar sets. A typical example of our jumptype processes is the symmetric jump process with jumping intensity e −c0(x,y)x−y � α2 α1 c(α, x, y) ν(dα) x − y  d+α where ν is a probability measure on [α1, α2] ⊂ (0, 2), c(α, x, y) is a jointly measurable function that is symmetric in (x, y) and is bounded between two positive constants, and c0(x, y) is a
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 32 (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.
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.
Manifolds and Graphs With Slow Heat Kernel Decay
 Invent. Math
, 1999
"... We give upper estimates on the long time behaviour of the heat kernel on a noncompact Riemannian manifold and infinite graphs, which only depend on a lower bound of the volume growth. We also show that these estimates are optimal. ..."
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Cited by 26 (2 self)
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We give upper estimates on the long time behaviour of the heat kernel on a noncompact Riemannian manifold and infinite graphs, which only depend on a lower bound of the volume growth. We also show that these estimates are optimal.
The Art of Random Walks
 Lecture Notes in Mathematics 1885
, 2006
"... 1.1 Basic definitions and preliminaries................ 8 ..."
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Cited by 15 (5 self)
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1.1 Basic definitions and preliminaries................ 8
Evolving sets, mixing and heat kernel bounds
 Probab. Theory Rel. Fields
, 2005
"... We show that a new probabilistic technique, recently introduced by the first author, yields the sharpest bounds obtained to date on mixing times of Markov chains in terms of isoperimetric properties of the state space (also known as conductance bounds or Cheeger inequalities). We prove that the boun ..."
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Cited by 10 (2 self)
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We show that a new probabilistic technique, recently introduced by the first author, yields the sharpest bounds obtained to date on mixing times of Markov chains in terms of isoperimetric properties of the state space (also known as conductance bounds or Cheeger inequalities). We prove that the bounds for mixing time in total variation obtained by Lovász and Kannan, can be refined to apply to the maximum relative deviation p n (x, y)/π(y) − 1  of the distribution at time n from the stationary distribution π. We then extend our results to Markov chains on infinite state spaces and to continuoustime chains. Our approach yields a direct link between isoperimetric inequalities and heat kernel bounds; previously, this link rested on analytic estimates known as Nash inequalities. 1
Mixing for Markov Chains and Spin Systems
 LECTURES GIVEN AT THE 2005 PIMS SUMMER SCHOOL IN PROBABILITY HELD AT THE UNIVERSITY OF BRITISH COLUMBIA FROM JUNE 6 THROUGH JUNE 30.
, 2005
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ASYMPTOTIC ENTROPY AND GREEN SPEED FOR RANDOM WALKS ON COUNTABLE GROUPS
, 2008
"... We study asymptotic properties of the Green metric associated with transient random walks on countable groups. We prove that the rate of escape of the random walk computed in the Green metric equals its asymptotic entropy. The proof relies on integral representations of both quantities with the exte ..."
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Cited by 4 (1 self)
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We study asymptotic properties of the Green metric associated with transient random walks on countable groups. We prove that the rate of escape of the random walk computed in the Green metric equals its asymptotic entropy. The proof relies on integral representations of both quantities with the extended Martin kernel. In the case of finitely generated groups, where this result is known (Benjamini and Peres [Probab. Theory Related Fields 98 (1994) 91–112]), we give an alternative proof relying on a version of the socalled fundamental inequality (relating the rate of escape, the entropy and the logarithmic volume growth) extended to random walks with unbounded support. 1. Introduction. Let Γ be an infinite countable group and let (Zn) be a transient random walk on Γ. In order to study asymptotic properties of the random walk, we define the Green (or hitting) metric,