On stochastic completeness of jump processes

Abstract

We prove the following sufficient condition for stochastic completeness of symmetric jump processes on metric measure spaces: if the volume of the metric balls grows at most exponentially with radius and if the distance function is adapted in a certain sense to the jump kernel then the process is stochastically complete. We use this theorem to prove the following criterion for stochastic completeness of a continuous time random walk on a graph with a counting measure: if the volume growth with respect to the graph distance is at most cubic then the random walk is stochastically complete, where the cubic volume growth is sharp.

Stochastic completeness for graphs with curvature dimension conditions

“…This can be circumvented by using intrinsic metrics introduced by Frank, Lenz and Wingert [FLW14], see e.g. [GHM12,Fol14,Hua14].…”

Conservation property of symmetric jump-diffusion processes

“…In [12,18,19], the "small jump" part E .1/ corresponds to finite range jumps and the "big jump" part E .2/ does to the remaining jumps. A point in our argument is that we let E .1/ correspond to relatively "small jumps" determined by some function F (see Assumption 2.2 below).…”

“…In order to prove Theorem 2.5, we use the so-called Davies method ( [5,9]) in a similar way to [12,18,19]. More precisely, we first divide the regular Dirichlet form .E; F / into the "small jump" part E .1/ and the "big jump" part E .2/ (see Section 2 for definition).…”

“…Motivated by the recent progress of the analysis of jump processes (see, e.g., [2] and references therein), there also have been papers on the conservation property of symmetric jump(-diffusion) processes generated by regular Dirichlet forms (see [12,18,19,23] and also [22,26] for related results). These papers characterize the conservation property in terms of either of the same factors as for diffusion processes: in [12,18,19], the volume of the underlying measure is allowed to grow exponentially, but the canonical coefficients are assumed to be bounded.…”

“…These papers characterize the conservation property in terms of either of the same factors as for diffusion processes: in [12,18,19], the volume of the underlying measure is allowed to grow exponentially, but the canonical coefficients are assumed to be bounded. In contrast with this, we allow in [23] the coefficients to be unbounded under the condition that the state space is R d ; however, to make use of the conservativeness criterion in terms of the (L 2 -)generator obtained by Isozaki and Uemura ( [15]), we assume there that the underlying measure is the Lebesgue measure on R d .…”