Rates of Convergence to Stationarity for Reflected Brownian Motion

We provide the first rate of convergence to stationarity analysis for reflected Brownian motion (RBM) as the dimension grows under some uniformity conditions. In particular, if the underlying routing matrix is uniformly contractive, uniform stability of the drift vector holds, and the variances of the underlying Brownian motion (BM) are bounded, then we show that the RBM converges exponentially fast to stationarity with a relaxation time of order $O\left(d^4{\left(\text{l}\text{o}\text{g}\left(d\right)\right)}^3\right)$ as the dimension d → ∞. Our bound for the relaxation time follows as a corollary of the nonasymptotic bound we obtain for the initial transient effect, which is explicit in terms of the RBM parameters.