Heat Kernel Estimates for Markov Process

Abstract

In this talk, we first discuss Markov process which is a stochastic process contained Brownian motion, especially in relation to a transition density which is also considered as a heat kernel. Then we consider the following two classes of Markov process: (1) the pure jump Markov process X whose jumping kernel is comparable to the measurable function with a weak scaling condition; (2) for d–independent 1–dimensional α–stable processes Yi , let Y := (Y1, . . . , Yd) be an anisotropic L?evy process. Then we consider anisotropic Markov process Z := (Z1, . . . , Zd) whose jumping kernel is comparable to that of Y . Finally, I will introduce the result on the sharp two-sided estimates of the transition densities for X and Z. The first project is joint work with Tomasz Grzywny and Panki Kim, and the second project is joint work with Moritz Kassmann and Takashi Kumagai. ? References [1] T. Grzywny, K.-Y. Kim and P. Kim Estimates of Dirichlet heat kernel for symmetric Markov processes. Stochastic Process. Appl. 130(1): 431–470, 2020. [2] M. Kassmann, K.-Y. Kim and T. Kumagai. Heat kernel bounds for nonlocal operators with singular kernels. 2019+. https://arxiv.org/abs/1910.04242.