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.
Publié le : 2002-10-14
Classification:
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,
60K37,
60K35,
60J80,
60G57,
60F05
@article{1039548380,
author = {Dawson, Donald A. and Fleischmann, Klaus and M\"orters, Peter},
title = {Strong clumping of super-Brownian motion in a stable catalytic medium},
journal = {Ann. Probab.},
volume = {30},
number = {1},
year = {2002},
pages = { 1990-2045},
language = {en},
url = {http://dml.mathdoc.fr/item/1039548380}
}
Dawson, Donald A.; Fleischmann, Klaus; Mörters, Peter. Strong clumping of super-Brownian motion in a stable catalytic medium. Ann. Probab., Tome 30 (2002) no. 1, pp. 1990-2045. http://gdmltest.u-ga.fr/item/1039548380/