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

First Online: 25 April 2007 Received: 18 July 2006 Revised: 07 March 2007 DOI :
10.1007/s00440-007-0070-5

Cite this article as: Chen, Z. & Kumagai, T. Probab. Theory Relat. Fields (2008) 140: 277. doi:10.1007/s00440-007-0070-5
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 γ_{1} and γ_{2} , where either γ_{2} ≥ γ_{1} > 0 or γ_{1} = γ_{2} = 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.

Mathematics Subject Classification (2000) Primary: 60J75 60J35 Secondary: 31C25 31C05 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.

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