| Summary: | 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 d<sub>s</sub><4, and prove that (2-d<sub>s</sub>/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.
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