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...

Let U be an open set in Rd. We show that under a mild assumption on the richness of the generator, a Feller process in (U,B(U)) with (predictable) killing is a semimartingale. To this end, we generalize the notion of semimartingales in a natural way to those ‘with killing’. Furthermore we...