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October 2002 Strong clumping of super-Brownian motion in a stable catalytic medium
Donald A. Dawson, Klaus Fleischmann, Peter Mörters
Ann. Probab. 30(4): 1990-2045 (October 2002). DOI: 10.1214/aop/1039548380


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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Donald A. Dawson. Klaus Fleischmann. Peter Mörters. "Strong clumping of super-Brownian motion in a stable catalytic medium." Ann. Probab. 30 (4) 1990 - 2045, October 2002.


Published: October 2002
First available in Project Euclid: 10 December 2002

zbMATH: 1029.60088
MathSciNet: MR1944014
Digital Object Identifier: 10.1214/aop/1039548380

Primary: 60J80 , 60K35 , 60K37
Secondary: 60F05 , 60G57

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

Rights: Copyright © 2002 Institute of Mathematical Statistics


Vol.30 • No. 4 • October 2002
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