Quenched convergence of a sequence of superprocesses in R^d among Poissonian obstacles

Details Quenched convergence of a sequence of superprocesses in R^d among Poissonian obstacles Quenched convergence of a sequence of superprocesses in R^d among Poissonian obstacles

2008-01-22

arxiv.org/abs/0801.3444v3

Stochastic Process. Appl., 119: 2598-2624, 2009

Mathematics

Probability

22 pages

Scientific paper

10.1016/j.spa.2009.01.004

We prove a convergence theorem for a sequence of super-Brownian motions moving among hard Poissonian obstacles, when the intensity of the obstacles grows to infinity but their diameters shrink to zero in an appropriate manner. The superprocesses are shown to converge in probability for the law $\mathbf{P}$ of the obstacles, and $\mathbf{P}$-almost surely for a subsequence, towards a superprocess with underlying spatial motion given by Brownian motion and (inhomogeneous) branching mechanism $\psi(u,x)$ of the form $\psi(u,x)= u^2+ \kappa(x)u$, where $\kappa(x)$ depends on the density of the obstacles. This work draws on similar questions for a single Brownian motion. In the course of the proof, we establish precise estimates for integrals of functions over the Wiener sausage, which are of independent interest.

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