Stable-like processes with indices greater than two

In Euclidean space, symmetric stable process with index \(\alpha \in (0,2)\) is obtained by a time change of Brownian motion using the subordinator of index \( \alpha/2 \)‚Äč. By a similar subordination, there exists stable-like random walks with indices greater than two on various fractals and fractal-like graphs. However, existing methods to prove transition probability estimates fail in this setting.

Davies' method is a technique introduced by E. B. Davies to obtain heat kernel upper bounds for uniformly elliptic operators in Euclidean space. This method was extended to general symmetric Markov semigroups by Carlen, Kusuoka and Stroock. In this talk, I will introduce some of the ideas involved in extending Davies method to obtain heat kernel bounds for stable-like processes with indices greater than two.

This talk is based on joint works with Laurent Saloff-Coste.