Heat kernel estimates for anomalous heavy-tailed random walks

Abstract

Sub-Gaussian estimates for the natural random walk is typical of many regular fractal graphs. Subordination shows that there exist heavy tailed jump processes whose jump indices are greater than or equal to two. However, the existing machinery used to prove heat kernel bounds for such heavy tailed random walks fail in this case. In this work we extend Davies' perturbation method to obtain transition probability bounds for these anomalous heavy tailed random walks. We prove global upper and lower bounds on the transition probability density that are sharp up to constants. An important feature of our work is that the methods we develop are robust to small perturbations of the symmetric jump kernel.Résumé. Pour de nombreux graphes réguliers de type fractal, la marche aléatoire simple satisfait de estimation de type sous-Gaussiennes. La technique de la subordination montre alors qu'il existe the processus tête type stage dont l'indice des sauts est supérieur ouégale a 2. Pour de telles processus, les techniques usuelles pour les estimations loin de la diagonale ne fonctionnent pas. Nouś etendons la célèbre méthode de Davies dans le cas de ces processusà sauts "anormaux". Nous obtenons des bornes supérieures et infrieures précises sur le noyaux de transition pas des méthodes qui sont stables sous de petites perturbations des sauts.

Heat kernel estimates for symmetric jump processes with mixed polynomial growths

“…One of major problems in this field is to obtain heat kernel estimates for jump processes on metric measure space without the restriction β 2 < 2. Recently, several articles have discussed this problem ( [3,22,30,38,49,52]). In [22], the authors established the stability results on heat kernel estimates of the form in (1.3) for symmetric jump Markov processes on metric measure spaces satisfying volume doubling and reverse volume doubling properties.…”

Decay of harmonic functions for discrete time Feynman--Kac operators with confining potentials

“…This condition includes many important examples of long-range random walks, for instance stable-like random walks in the integer lattice, see e.g. Bass and Levin [7], as well as random walks in measure metric spaces studied recently by Murugan and Saloff-Coste in [48] and [49]. In Corollary 4.2 we also extend this observation to kernels with much lighter tails.…”

The Davies method revisited for heat kernel upper bounds of regular Dirichlet forms on metric measure spaces

“…To prove the off-diagonal upper bound (UE), we need the following two lemmas. We begin with the first one, Lemma 3.4 below, which can be proved as in [31,Lemma 3.7], see also [9,Lemma 3.21]. Lemma 3.4.…”

“…Recently, Murugan and Saloff-Coste extend the Davies method developed in [9,13] and obtain heat kernel upper bounds, for local Dirichlet forms on metric spaces in [32] and for non-local Dirichlet forms on infinite graphs in [31], where a cutoff inequality introduced in [1] plays an important role.…”

“…These characterization are stable under bounded perturbation of Dirichlet forms. We mention that one of our starting point here is from the cutoff inequality on balls to be stated below, labelled by condition (CIB), which is subtly distinct from the similar conditions in previous papers [1], [23], [32,31], [12], [16]. Also the metric space considered in this paper is allowed to be bounded or unbounded, unlike the most previous ones in which the metric space is always assumed to be unbounded.…”