Strong clumping of super-Brownian motion in a stable catalytic medium
- Dawson, Donald A.
- Fleischmann, Klaus
- Mörters, Peter
2010 Mathematics Subject Classification
- 60K37 60K35 60J80 60G57 60F05
- 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
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.
- Ann. Probab. 30(4) (2002), pp. 1990-2045