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Sub-Gaussian estimates of heat kernels on infinite graphs
- Duke Math. J
, 2000
"... We prove that a two sided sub-Gaussian 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 27 (8 self)
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We prove that a two sided sub-Gaussian 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 sub-Gaussian estimates for random walks
- Math. Annalen
, 2002
"... We show that a fi-parabolic Harnack inequality for random walks on graphs is equivalent, on one hand, to so called fi-Gaussian 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 24 (4 self)
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We show that a fi-parabolic Harnack inequality for random walks on graphs is equivalent, on one hand, to so called fi-Gaussian 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 non-compact 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 19 (2 self)
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We give upper estimates on the long time behaviour of the heat kernel on a non-compact Riemannian manifold and infinite graphs, which only depend on a lower bound of the volume growth. We also show that these estimates are optimal.
Asymptotic separation for independent trajectories of Markov processes
"... Markov processes 15 7 Examples 18 7.1 Diffusions on manifolds : : : : : : : : : : : : : : : : : : : : : : : : : : 19 7.2 The ff-process : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 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 ff-process : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 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
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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Cited by 1 (1 self)
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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
