Dimension-free local convergence and perturbations for reflected Brownian motions

Abstract

We describe and analyze a class of positive recurrent reflected Brownian motions (RBMs) in ${\mathbb{R}}_{+}^{\mathit{d}}$ 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.