Dimension-free local convergence and perturbations for reflected Brownian motions

TL;DR

This paper establishes dimension-free local convergence rates for a class of reflected Brownian motions in high dimensions, using contraction estimates and analyzing specific models like the Symmetric Atlas model.

Contribution

It introduces conditions under which high-dimensional RBMs exhibit dimension-independent convergence rates and analyzes the Symmetric Atlas model for polynomial convergence.

Findings

Dimension-independent stretched exponential convergence rates for certain RBMs.

Polynomial convergence rates for the gap process in the Symmetric Atlas model.

Analysis of pathwise derivatives linking to random walks in random environments.

Abstract

We describe and analyze a class of positive recurrent reflected Brownian motions (RBMs) in $\mathbb{R}^d_+$ for which local statistics converge to equilibrium at a rate independent of the dimension $d$. Under suitable assumptions on the reflection matrix, drift and diffusivity coefficients, dimension-independent stretched exponential convergence rates are obtained by estimating contractions in an underlying weighted distance between synchronously coupled RBMs. We also study the Symmetric Atlas model as a first step in obtaining dimension-independent convergence rates for RBMs not satisfying the above assumptions. By analyzing a pathwise derivative process and connecting it to a random walk in a random environment, we obtain polynomial convergence rates for the gap process of the Symmetric Atlas model started from appropriate perturbations of stationarity.