Reflected Brownian Motion in a Convex Polyhedral Cone: Tail Estimates for the Stationary Distribution

TL;DR

This paper establishes conditions for the existence, uniqueness, and exponential convergence of the stationary distribution of multidimensional reflected Brownian motion in convex cones, providing tail estimates and applications to rank-based particle systems.

Contribution

It introduces new sufficient conditions for stationary distribution existence and exponential convergence, along with tail estimates, for reflected Brownian motions in convex polyhedral cones.

Findings

Existence of stationary distribution under certain conditions

Exponential convergence to the stationary distribution

Finite exponential moments for the stationary distribution

Abstract

Consider an multidimensional obliquely reflected Brownian motion in the positive orthant, or, more generally, in a convex polyhedral cone. We find sufficient conditions for existence of a stationary distribution and convergence to this distribution at the exponential rate, as time goes to infinity. We also prove that certain exponential moments for this distribution are finite, thus providing a tail estimate for this distribution. Finally, we apply these results to systems of rank-based competing Brownian particles.