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.

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Annals of Probability
School of Mathematics and Statistics

Dawson, D.A, Fleischmann, K. (Klaus), & Mörters, P. (Peter). (2002). Strong clumping of super-Brownian motion in a stable catalytic medium. Annals of Probability, 30(4), 1990–2045. doi:10.1214/aop/1039548380