The Annals of Probability

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

Donald A. Dawson, Klaus Fleischmann, and Peter Mörters

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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.

Article information

Source
Ann. Probab., Volume 30, Number 4 (2002), 1990-2045.

Dates
First available in Project Euclid: 10 December 2002

Permanent link to this document
https://projecteuclid.org/euclid.aop/1039548380

Digital Object Identifier
doi:10.1214/aop/1039548380

Mathematical Reviews number (MathSciNet)
MR1944014

Zentralblatt MATH identifier
1029.60088

Subjects
Primary: 60K37: Processes in random environments 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Secondary: 60G57: Random measures 60F05: Central limit and other weak theorems

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

Citation

Dawson, Donald A.; Fleischmann, Klaus; Mörters, Peter. Strong clumping of super-Brownian motion in a stable catalytic medium. Ann. Probab. 30 (2002), no. 4, 1990--2045. doi:10.1214/aop/1039548380. https://projecteuclid.org/euclid.aop/1039548380


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  • OTTAWA, ONTARIO K1S 5B6 CANADA E-MAIL: ddawson@math.carleton.ca K. FLEISCHMANN WEIERSTRASS INSTITUTE FOR APPLIED ANALy SIS AND STOCHASTICS MOHRENSTR. 39 D-10117 BERLIN GERMANY E-MAIL: fleischmann@wias-berlin.de P. MÖRTERS DEPARTMENT OF MATHEMATICAL SCIENCES UNIVERSITY OF BATH CLAVERTON DOWN BATH BA2 7AY UNITED KINGDOM E-MAIL: P.Moerters@maths.bath.ac.uk