Conditional survival distributions of Brownian trajectories in a one dimensional Poissonian environment in the critical case

TL;DR

This paper investigates the behavior of a one-dimensional Brownian motion with drift in a Poissonian obstacle environment, focusing on the critical case where drift equals obstacle intensity, and establishes convergence of survival-conditioned processes.

Contribution

It extends previous work by analyzing the critical case where the drift matches the obstacle intensity, proving convergence of the process law conditioned on survival.

Findings

Convergence of the law of processes conditioned on survival as time tends to infinity.

Complements previous results for the subcritical case where drift is less than obstacle intensity.

Provides insights into the critical regime of Brownian motion in random environments.

Abstract

In this work we consider a one-dimensional Brownian motion with constant drift moving among a Poissonian cloud of obstacles. Our main result proves convergence of the law of processes conditional on survival up to time $t$ as $t$ converges to infinity in the critical case where the drift coincides with the intensity of the Poisson process. The complements a previous result of T. Povel, who considered the same question in the case where the drift is strictly smaller than the intensity.