In this paper, we establish an oscillation estimate of nonnegative harmonic functions for a pure-jump subordinate Brownian motion. The infinitesimal generator of such subordinate Brownian motion is an integro-differential operator. As an application, we give a probabilistic proof of the...

In this paper, we consider a large class of subordinate Brownian motions X via subordinators with Laplace exponents which are complete Bernstein functions satisfying some mild scaling conditions at zero and at infinity. We first discuss how such conditions govern the behavior of the subordinator...

In this paper, we study sharp Dirichlet heat kernel estimates for a large class of symmetric Markov processes in C1,η open sets. The processes are symmetric pure jump Markov processes with jumping intensity κ(x,y)ψ1(|x−y|)−1|x−y|−d−α, where α∈(0,2). Here, ψ1 is an increasing...