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...

For α∈(0,2) and M0, we consider a family of nonlocal operators {Δ+aαΔα/2,a∈(0,M]} on Rd under Kato class gradient perturbation. We establish the existence and uniqueness of their fundamental solutions, and derive their sharp two-sided estimates. The estimates give explicit dependence on...

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...