WIAS Preprint No. 331, (1997)

Smooth density field of catalytic super-Brownian motion



Authors

  • Fleischmann, Klaus
  • Klenke, Achim

2010 Mathematics Subject Classification

  • 60J80 60G57 60K35

Keywords

  • catalytic super-Brownian medium, catalyst, superprocess, measure-valued branching, non-extinction, persistence, two-dimensional process, equilibrium state, absolutely continuous states, self-similarity, time-space gaps of super-Brownian motion, asymptotic density, Cinfinity - density field, heat solution, Brownian excursions, occupation density, super-local time

Abstract

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.

Appeared in

  • Ann. Appl. Probab. 9(2) (1999), pp. 298-318

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