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 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, dimensionindependent 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.

Domains of attraction of invariant distributions of the infinite Atlas model

“…Further, under very general stability and nondegeneracy conditions on the drift and diffusion coefficients {a j }, {b j }, the RBM describing gaps between the ranked particles has a unique stationary measure [HW87a]. Geometric ergodicity results can be found in [BL07] and rates of convergence to stationarity, depending explicitly on {a j }, {b j } have recently been obtained [BB20b,BB20a].…”