Non-local operators, jump diffusions and Feynman-Kac tranforms

Abstract

Non-local operators are analytically defined by integrals over the whole space, hence hard to study certain properties. This thesis studies inverse local times at $0$ of one-dimensional reflected diffusions on $[0, \infty)$, and establishes a new comparison principle for inverse local times. As an application, we obtain the Green function estimates for a class of non-local operators.\\ We further study diffusions with jumps, which are associated with the combination of local and non-local operators. We show that the two-sided heat kernel estimates for a class of (not necessarily symmetric) diffusions with jumps are stable under non-local Feynman-Kac perturbations.