WIAS Preprint No. 305, (1996)

Convergence to a non-trivial equilibrium for two-dimensional 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, local L^2-Lipschitz continuity

Abstract

In contrast to the classical super-Brownian motion (SBM), the SBM (Xϱt) t ≥ 0 in a super-Brownian medium ϱ (constructed in [DF96a]) is known to be persistent in all three dimensions of its non-trivial existence: The full intensity is carried also by all longtime limit points ([DF96a, DF96b, EF96]). Uniqueness of the accumulation point, however, has been shown so far only in dimensions d=1 and d=3 ([DF96a, DF96b]). Here we fill this gap and show that convergence also holds in the critical dimension d=2. We identify the limit as a random multiple of Lebesgue measure.

Our main tools are a self-similarity of Xϱ in d=2 and the fact that the medium has "gaps" in the space-time picture. The self-similarity implies that persistent convergence of Xϱt as t → ∞ is equivalent to the absolute continuity of Xϱt at a fixed time t > 0. Absolute continuity however will be obtained via the fact that in absence of the catalytic medium, Xϱ is smoothed according to the heat flow.

Appeared in

  • Ann. Appl. Probab., 9(2) (2000), pp. 298-318, under new title: Smooth density field of catalytic super-Brownian motion.

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