Stability of heat kernel estimates and parabolic Harnack inequalities for jump processes on metric measure spaces

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 Poincaré 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).