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

Abstract

A typical feature of the long time behavior 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 theorem under renormalization. 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 non-Markovian 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.