Dimension-free local convergence and perturbations for reflected Brownian motions

2020/09/27 by Banerjee, Sayan, Brown, Brendan
#37A25 #60J55 #60J60 #60K37 #FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.2009.12937

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, 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.

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