Probability Seminar: Takashi Kumagai

We consider mixed-type jump processes on metric measure spaces and prove the stability of two-sided heat kernel estimates, heat kernel upper bounds, and parabolic  Harnack inequalities. We establish their stable equivalent characterizations in terms of the jump kernels, modifications of cut-off Sobolev inequalities, and the Poincar\'e inequalities. In particular, we prove the stability of heat kernel estimates for $\alpha$-stable-like processes even with $\alpha\ge 2$, which has been one of the major open problems in this area. We will also explain applications to stochastic processes on fractals.  

This is a joint work with Z.Q. Chen (Seattle) and J. Wang (Fuzhou).