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Beschreibung:
Given an (ordinary) super-Brownian motion (SBM) ϱ on Rd of dimension d = 2, 3, we consider a (catalytic) SBM Xϱ on Rd with "local branching rates" ϱs(dx). We show that Xϱt is absolutely continuous with a density function ξϱt, say. Moreover, there exists a version of the map (t,z) ↦ ξϱt(z) which is C∞ and solves the heat equation off the catalyst ϱ, more precisely, off the (zero set of) closed support of the time-space measure ds ϱs(dx). Using self-similarity, we apply this result to answer the question of the long-term behavior of Xϱ in dimension d = 2 : If ϱ and Xϱ start with a Lebesgue measure, then XϱT converges (persistently) as T → ∞ towards a random multiple of Lebesgue measure.