, Volume 140, Issue 1-2, pp 277-317
Date: 25 Apr 2007

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

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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.

Dedicated to Professor Masatoshi Fukushima on the occasion of his 70th birthday.
The research of Zhen-Qing Chen is supported in part by NSF Grants DMS-0303310 and DMS-06000206. The research of Takashi Kumagai is supported in part by the Grant-in-Aid for Scientific Research (B) 18340027.