Parabolic Harnack inequality and heat kernel estimates for random walks with long range jumps

By Martin T. Barlow, Richard F. Bass, Takashi Kumagai. 2007

57 citations recorded by the source corpus.

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.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/d2adef0a3a4674f083a75f26e80778a710f4a29a

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In Applied Mathematics

In Geometry and Topology

Citations in the corpus

Stability of parabolic Harnack inequalities for symmetric non-local Dirichlet forms (2016, methodology): “t describes how the process Xjumps) to connect PHI(φ) with the properties of the associated heat kernel and jumping kernel. For instance, UJS yielded by a probabilistic consideration and motivated by [BBK2] in a graph setting plays one of key roles for the characterization of PHI(φ) in the present f”
A priori Holder estimate, parabolic Harnack principle and heat kernel estimates for diffusions with jumps (2008, background): “14) was introduced in [3, 7] and it was denoted as (UJS)≤1 there.”
On Harnack Inequality and Hölder Regularity for Isotropic Unimodal Lévy Processes (2013, result): “Note that similar condition as (A1) appeared in [4] and [1] and exactly the same in [5] and [6].”
Regularity results for nonlocal parabolic equations (2013, background): “neglected resp. not treated according to their presence in journals. One such area is given by heat kernel bounds for more general nonlocal Dirichlet forms on general state spaces. The articles [22], [3], [17] extend [21] and treat regularity and estimates of the heat kernel. For a general introduct”
Weighted Poincaré inequality and heat kernel estimates for finite range jump processes (2008, methodology): “Our proof uses a combination of the techniques developed in [CK1, CK2] and in [ BBK , BBCK].”