Heat kernel estimates for jump processes of mixed types on metric measure spaces

Abstract

In this paper, we investigate symmetric jump-type processes on a class of metric measure spaces with jumping intensities comparable to radially symmetric functions on the spaces. The class of metric measure spaces includes the Alfors d-regular sets, which is a class of fractal sets that contains geometrically self-similar sets. A typical example of our jump-type processes is the symmetric jump process with jumping intensity $$e^{-c_0 (x, y)|x-y|}\, \int_{\alpha_1}^{\alpha_2} \frac{c(\alpha, x, y)}{|x-y|^{d+\alpha}} \, \nu (d\alpha)$$ where ν is a probability measure on $$[\alpha_1, \alpha_2]\subset (0, 2)$$ , c(α, x, y) is a jointly measurable function that is symmetric in (x, y) and is bounded between two positive constants, and c 0(x, y) is a jointly measurable function that is symmetric in (x, y) and is bounded between γ1 and γ2, where either γ2 ≥ γ1 > 0 or γ1 = γ2 = 0. This example contains mixed symmetric stable processes on $${\mathbb{R}}^n$$ as well as mixed relativistic symmetric stable processes on $${\mathbb{R}}^n$$ . We establish parabolic Harnack principle and derive sharp two-sided heat kernel estimate for such jump-type processes.