Dimension results and local times for superdiffusions on fractals

Ben Hambly and Peter Koepernik

Stochastic Processes and their Applications, 2023, vol. 158, issue C, 377-417

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.

Keywords: Superprocesses; Hausdorff dimension; Local times; Diffusion processes; Fractal; Moments (search for similar items in EconPapers)
Date: 2023
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DOI: 10.1016/j.spa.2023.01.008

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