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Dimension results and local times for superdiffusions on fractals

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  • Hambly, Ben
  • Koepernik, Peter

Abstract

We consider the Dawson–Watanabe superprocess obtained from a spatial motion with sub-Gaussian transition densities on a metric measure space with finite Hausdorff dimension, and examine the dimensions of the range and the set of times when the support intersects a given set, generalising results of Serlet and Tribe. As intermediate results, we prove existence of local times for the superprocess if the spectral dimension of the spatial motion satisfies ds<4, and prove that (2−ds/2)∧1 is the critical Hölder-continuity exponent in the time variable. Furthermore, we prove a bound on moments of the integrated superprocess, and give uniform upper bounds on the mass the superprocess assigns to small balls, generalising a result of Perkins.

Suggested Citation

  • Hambly, Ben & Koepernik, Peter, 2023. "Dimension results and local times for superdiffusions on fractals," Stochastic Processes and their Applications, Elsevier, vol. 158(C), pages 377-417.
    • Handle: RePEc:eee:spapps:v:158:y:2023:i:c:p:377-417
    DOI: 10.1016/j.spa.2023.01.008
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    References listed on IDEAS

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    1. Adler, Robert J. & Lewin, Marica, 1992. "Local time and Tanaka formulae for super Brownian and super stable processes," Stochastic Processes and their Applications, Elsevier, vol. 41(1), pages 45-67, May.
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