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

TL;DR

This paper proves that super-Brownian motions among dense Poissonian obstacles converge to a superprocess with a modified branching mechanism, with key estimates on Wiener sausages aiding the proof.

Contribution

It establishes a convergence theorem for super-Brownian motions in obstacle fields and introduces new estimates for Wiener sausage integrals.

Findings

Superprocesses converge in probability and almost surely to a limit process.

The limiting superprocess has a spatially dependent branching mechanism.

New integral estimates for Wiener sausages are developed.

Abstract

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.