# Parameter and dimension dependence of convergence rates to stationarity for Reflecting Brownian Motions

## TL;DR

This paper derives explicit convergence rates to stationarity for multidimensional reflected Brownian motions, improving existing bounds and providing new dimension-dependent estimates applicable to various rank-based diffusions.

## Contribution

It provides the first explicit dimension and parameter-dependent convergence rates for RBMs and rank-based diffusions under Wasserstein distance, without requiring stationarity or reversibility.

## Key findings

- Improved relaxation time bounds from O(d^4(log d)^2) to O((log d)^2) for certain RBMs.
- First explicit rates for rank-based diffusions including the Atlas model.
- Relaxation time bound of O(d^6(log d)^2) for the standard Atlas model.

## Abstract

We obtain rates of convergence to stationarity in L^1-Wasserstein distance for a d-dimensional reflected Brownian motion (RBM) in the nonnegative orthant that are explicit in the dimension and the system parameters. The results are then applied to a class of RBMs considered in Blanchet-Chen (2016) and to rank-based diffusions including the Atlas model. In both cases, we obtain explicit rates and bounds on relaxation times. In the first case we improve the relaxation time estimates of O(d^4(log d)^2) obtained in Blanchet-Chen (2016) to O((log d)^2). In the latter case, we give the first results on explicit parameter and dimension dependent rates under the Wasserstein distance. The proofs do not require an explicit form for the stationary measure or reversibility of the process with respect to this measure, and cover settings where these properties are not available. In the special case of the standard Atlas model, we obtain a bound on the relaxation time of O(d^6(log d)^2).