Jump Processes on Spaces With Non-Uniform Volume Growth

Abstract

Abstract. Non-local Dirichlet forms with appropriately chosen jump kernels are used to define Markov pure jump processes on metric measure spaces that do not necessarily possess uniform volume growth. They may be seen as generalizations of stable (or stable-like) processes. Scaling properties and estimates on mean exit and hitting times are established. For some cases they provide enough information to conclude the continuity of related harmonic functions. Typical ultracontractivity arguments entail the existence of transition densities; their joint continuity is then deduced from the preceding results.