Worst-case mixing estimates for Brownian motion with semipermeable barriers

TL;DR

This paper analyzes the mixing times of Brownian motion constrained by semipermeable barriers in planar domains, providing bounds that decay exponentially with domain size, highlighting the worst-case behavior.

Contribution

It establishes exponential bounds on mixing times and stationary distribution lower bounds for Brownian motion with barriers, a novel analysis of worst-case mixing behavior.

Findings

Upper bound on mixing time in terms of geometric parameters

Lower bound on stationary distribution in worst-case scenarios

Exponential decay of bounds as domain size increases

Abstract

We study the mixing properties of a Brownian motion whose movements are hindered by semipermeable barriers. Our setting assumes that the process takes values in a smooth planar domain and that the barriers are one-dimensional closed curves. We establish an upper bound on the mixing time and a lower bound on the stationary distribution in terms of geometric parameters. These worst-case bounds decay at an exponential rate as the domain grows large, and we give examples that show that exponential decay is necessary in our worst-case setting.