Escape probabilities for branching Brownian motion among mild obstacles

TL;DR

This paper analyzes the probability of a critical branching Brownian motion escaping a large domain amid mild Poissonian obstacles, using PDEs and homogenization techniques to derive asymptotic results.

Contribution

It introduces a novel approach combining PDE analysis and quenched homogenization to quantify escape probabilities in complex obstacle environments.

Findings

Derived asymptotics for escape probabilities in obstacle-laden environments.

Established a quenched homogenization theorem for branching Brownian motion.

Connected PDE solutions with probabilistic escape behaviors.

Abstract

We derive asymptotics for the quenched probability that a critical branching Brownian motion killed at a small rate in Poissonian obstacles exits a large domain. Results are formulated in terms of the solution to a semilinear partial differential equation with singular boundary conditions. The proofs depend on a quenched homogenization theorem for branching Brownian motion among mild obstacles.