WIAS Preprint No. 636, (2001)

Strong clumping of super-Brownian motion in a stable catalytic medium



Authors

  • Dawson, Donald A.
  • Fleischmann, Klaus
  • Mörters, Peter

2010 Mathematics Subject Classification

  • 60K37 60K35 60J80 60G57 60F05

Keywords

  • catalytic super-Brownian motion, stable catalysts, critical branching, measure-valued branching, random medium, clumping, functional limit law, historical superprocess, Brownian snake in a random medium, subordination, exit measures, good and bad paths, stopped measures, collision local time, heavy tails, Feynman-Kac formula, annealed and quenched random medium approach

Abstract

A typical feature of the long time behaviour of continuous super-Brownian motion in a stable catalytic medium is the development of large mass clumps or clusters at spatially rare sites. We describe this phenomenon by means of a functional limit law under renormalisation. The limiting process is a Poisson point field of mass clumps with no spatial motion component and with infinite variance. The mass of each cluster evolves independently according to a continuous process trapped at mass zero, which we describe explicitly by means of a Brownian snake construction in a random medium. We also determine the survival probability and asymptotic size of the clumps.

Appeared in

  • Ann. Probab. 30(4) (2002), pp. 1990-2045

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