Probability Theory and Related Fields

, Volume 140, Issue 1, pp 277–317

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

Authors

    • Department of MathematicsUniversity of Washington
  • Takashi Kumagai
    • Research Institute for Mathematical SciencesKyoto University
Article

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 c0(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.

Mathematics Subject Classification (2000)

Primary: 60J7560J35Secondary: 31C2531C05

Copyright information

© Springer-Verlag 2007