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Parabolic Harnack inequality and heat kernel estimates for random walks with long range jumps
, 2008
"... We investigate the relationships between the parabolic Harnack inequality, heat kernel estimates, some geometric conditions, and some analytic conditions for random walks with long range jumps. Unlike the case of diffusion processes, the parabolic Harnack inequality does not, in general, imply the c ..."
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Cited by 19 (8 self)
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We investigate the relationships between the parabolic Harnack inequality, heat kernel estimates, some geometric conditions, and some analytic conditions for random walks with long range jumps. Unlike the case of diffusion processes, the parabolic Harnack inequality does not, in general, imply the corresponding heat kernel estimates.
The Einstein relation for random walks on graphs
, 2008
"... This paper investigates the Einstein relation; the connection between the volume growth, the resistance growth and the expected time a random walk needs to leave a ball on a weighted graph. The Einstein relation is proved under different set of conditions. In the simplest case it is shown under the ..."
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Cited by 5 (1 self)
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This paper investigates the Einstein relation; the connection between the volume growth, the resistance growth and the expected time a random walk needs to leave a ball on a weighted graph. The Einstein relation is proved under different set of conditions. In the simplest case it is shown under the volume doubling and time comparison principles. This and the other set of conditions provide the basic vwork for the study of (sub) diffusive behavior of the random walks on weighted graphs. 1
Harnack Inequalities: an introduction
, 2007
"... The aim of this article is to give an introduction to certain inequalities named after Carl ..."
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Cited by 2 (0 self)
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The aim of this article is to give an introduction to certain inequalities named after Carl
Parabolic Harnack inequality and heat kernel estimates for random walks with long range jumps
, 2008
"... We investigate the relationships between the parabolic Harnack inequality, heat kernel estimates, some geometric conditions, and some analytic conditions for random walks with long range jumps. Unlike the case of diffusion processes, the parabolic Harnack inequality does not, in general, imply the ..."
Abstract
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We investigate the relationships between the parabolic Harnack inequality, heat kernel estimates, some geometric conditions, and some analytic conditions for random walks with long range jumps. Unlike the case of diffusion processes, the parabolic Harnack inequality does not, in general, imply the corresponding heat kernel estimates. 1
Parabolic Harnack inequality and heat kernel estimates for random walks with long range jumps
, 2007
"... We investigate the relationships between the parabolic Harnack inequality, heat kernel estimates, some geometric conditions, and some analytic conditions for random walks with long range jumps. Unlike the case of diffusion processes, the parabolic Harnack inequality does not, in general, imply the ..."
Abstract
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We investigate the relationships between the parabolic Harnack inequality, heat kernel estimates, some geometric conditions, and some analytic conditions for random walks with long range jumps. Unlike the case of diffusion processes, the parabolic Harnack inequality does not, in general, imply the corresponding heat kernel estimates. 1
Correction to “SubGaussian estimates of heat kernels on infinite graphs ” by A.Grigor’yan and A.Telcs
, 2004
"... the proof of the first implication (G) ⇒ (HG) contained an error. Despite that, the result (G) ⇒ (H) remains true, which is proved below using a modified definition of (HG). 10 The Harnack inequality and the Green kernel Recall that the weighted graph (Γ, µ) satisfies the elliptic Harnack inequali ..."
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the proof of the first implication (G) ⇒ (HG) contained an error. Despite that, the result (G) ⇒ (H) remains true, which is proved below using a modified definition of (HG). 10 The Harnack inequality and the Green kernel Recall that the weighted graph (Γ, µ) satisfies the elliptic Harnack inequality (H) if there exist constants H,K> 1 such that, for all z ∈ Γ, R ≥ 1, and for any nonnegative function u in B(z,KR) which is harmonic in B(z,KR), the following inequality is satisfied1 max B(z,R) u ≤ H min
Consider the additive group Z of all integers as a...
, 2004
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GREEN KERNEL ESTIMATES AND THE FULL MARTIN BOUNDARY FOR RANDOM WALKS ON LAMPLIGHTER GROUPS AND DIESTELLEADER GRAPHS
, 2004
"... Abstract. We determine the precise asymptotic behaviour (in space) of the Green kernel of simple random walk with drift on the DiestelLeader 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 DiestelLeader 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 DLgraphs 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.