Jump Diffusions and Non-local Operators

Transition density function encodes all the information about a Markov process. It is the fundamental solution (also called heat kernel) of the infinitesimal generator of the corresponding Markov process. When the Markov process is a diffusion on Euclidean space, its infinitesimal generator is a differential operator. When the Markov process has discontinuous sample paths, its infinitesimal generator is a non-local operator. Heat kernels for second order elliptic operators have been well studied and there are many beautiful results. The study of heat kernels for non-local operators is quite recent. In this talk, the speaker will survey the recent progress in the study of heat kernels of non-local operators, focusing on their two-sided estimates, stability, and their applications to solutions of stochastic differential equations driven by Levy processes.