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Heat kernel estimates for jump processes of mixed types on metric measure spaces
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  • Published: 25 April 2007

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

  • Zhen-Qing Chen1 &
  • Takashi Kumagai2 

Probability Theory and Related Fields volume 140, pages 277–317 (2008)Cite this article

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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.

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Authors and Affiliations

  1. Department of Mathematics, University of Washington, Seattle, WA, 98195, USA

    Zhen-Qing Chen

  2. Research Institute for Mathematical Sciences, Kyoto University, Kyoto, 606-8502, Japan

    Takashi Kumagai

Authors
  1. Zhen-Qing Chen
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  2. Takashi Kumagai
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Corresponding author

Correspondence to Zhen-Qing Chen.

Additional information

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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Chen, ZQ., Kumagai, T. Heat kernel estimates for jump processes of mixed types on metric measure spaces. Probab. Theory Relat. Fields 140, 277–317 (2008). https://doi.org/10.1007/s00440-007-0070-5

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  • Received: 18 July 2006

  • Revised: 07 March 2007

  • Published: 25 April 2007

  • Issue Date: January 2008

  • DOI: https://doi.org/10.1007/s00440-007-0070-5

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Mathematics Subject Classification (2000)

  • Primary: 60J75
  • 60J35
  • Secondary: 31C25
  • 31C05
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