A class of singular symmetric Markov processes

Abstract

We consider a class of pure jump Markov processes in [special characters omitted] whose jump kernels are comparable to that of a certain d-dimensional Lévy process. Upper and lower bounds for the transition densities of these processes are obtained. We show that bounded harmonic functions associated with these processes are Hölder continuous. We construct Markov chain approximations for our processes. We give the construction and prove properties of the approximating Markov chains, and give a condition for the weak convergence of Markov chains to our Markov processes when a certain parameter α ∈ [1/2,2).