Uniform Ergodicity for Brownian Motion in a Bounded Convex Set

TL;DR

This paper establishes new bounds on the uniform ergodicity rates of high-dimensional Brownian motion confined within convex sets, independent of boundary smoothness and ambient dimension.

Contribution

It provides the first dimension-independent bounds on ergodicity rates for Brownian motion in convex sets with fixed diameter.

Findings

Bounds are independent of boundary smoothness.

Bounds do not depend on ambient dimension n.

Applicable to convex sets with given diameter.

Abstract

We consider an n-dimensional Brownian Motion trapped inside a bounded convex set by normally-reflecting boundaries. It is well-known that this process is uniformly ergodic. However, the rates of this ergodicity are not well-understood, especially in the regime of very high-dimensional sets. Here we present new bounds on these rates for convex sets with a given diameter. Our bounds do not depend upon the smoothness of the boundary nor the value of the ambient dimension, n.