Harnack inequalities and sub-Gaussian estimates for random walks

By Alexander Grigorʼyan, András Telcs. 2002

119 citations recorded by the source corpus.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/8a6fce26ef01cb3cd9afd887e66266b8b6bb3db8

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In Applied Mathematics

In Geometry and Topology

Citations in the corpus

Heat Kernels on Metric Measure Spaces (2014, result): “7] ) is to obtain an annulus Harnack inequality for the Green function, without assuming any specific properties of the metric d, unlike previously known similar results in [6], [34] where the geodesic property of the distance function was used.”
Stability of parabolic Harnack inequalities (2003, background): “Theorem 1.2. ([ GT2 , Theorem 3.1.]) Let (G;E) be an innite connected graph with conductances a = (axy) satisfying the (p0) condition.”
Stability of parabolic Harnack inequalities on metric measure spaces (2006, background): “(See also [HSC] for the case where solutions to the heat equation are sucient regular.) That these are equivalent to (c) and (d) is proved in [GT3]. (See [ GT2 ] for the graph case.)”
Characterization of sub‐Gaussian heat kernel estimates on strongly recurrent graphs (2005, methodology): “One can then conclude that (HK(β)) holds by using the results in [30], but we will give a shorter proof by taking advantage of our strong recurrence assumption.”
TWO-SIDED ESTIMATES OF HEAT KERNELS ON METRIC MEASURE SPACES (2012, background): “12) was proved by the authors in [34], [35].”
Heat kernels and function theory on metric measure spaces (2003, background): “2) are valid on fractal-like graphs (see [6], [19], [20], [22], [30]).”
Stability of elliptic Harnack inequality (2016, background): “ar on each cable. Then his harmonic on D. Deﬁnition 6.1. We say that Ghas controlled weights if there exists p0 >0 such that wxy wx ≥ p0 for all x∈ V,y∼ x. (6.2) This is called the p0 condition in [GT]. Note that it implies that vertices have degree at most 1/p0, so that an unweighted graph satis”
Parabolic Harnack inequality and heat kernel estimates for random walks with long range jumps (2007, methodology): “In [ GT ] it is proved that for simple random walks the condition VD+EHI +E2 is also equivalent to conditions (a)‐(c) of Theorem 1.1.”
Local Nash Inequality and Inhomogeneity of Heat Kernels (2004, methodology): “For random walks on infinite graphs, Grigori’yan-Telcs[12] has shown that the sub-Gaussian heat kernel estimate”
Quenched invariance principles for walks on clusters of percolation or among random conductances (2004, background): “With the help of known Gaussian bounds on the transition kernel of the walk, cf. Delmotte [10], Grigoryan-Telcs [ 12 ], Barlow-Bass [4], the main task is to show that for any M> 0,”
On the equivalence of parabolic Harnack inequalities and heat kernel estimates (2012, methodology): “In this appendix, we modify the proof in [ GT2 ] and prove VD+EHI+RES( ) ) E( ) in a general MMD framework.”