Product-form invariant measures for Brownian motion with drift satisfying a skew-symmetry type condition

TL;DR

This paper extends reflected Brownian motion in polyhedral domains by incorporating a potential-dependent drift, demonstrating that the invariant measure retains a product form under a skew-symmetry condition, regardless of the potential.

Contribution

It generalizes the classical RBM invariant measure result to a broader class with potential-dependent drifts, establishing the skew-symmetry condition as essential.

Findings

Invariant measure in product form under skew-symmetry

Independence of potential in the skew-symmetry condition

Applicability to particle systems and rank-dependent diffusions

Abstract

Motivated by recent developments on random polymer models we propose a generalisation of reflected Brownian motion (RBM) in a polyhedral domain. This process is obtained by replacing the singular drift on the boundary by a continuous one which depends, via a potential U, on the position of the process relative to the domain. It was shown by Harrison and Williams (1987) that RBM in a polyhe- dral domain has an invariant measure in product form if a certain skew-symmetry condition holds. We show that (modulo technical assumptions) the generalised RBM has an invariant measure in product form if (and essentially only if) the same skew-symmetry condition holds, independent of the choice of potential. The invari- ant measure of course does depend on the potential. Examples include TASEP-like particle systems, generalisations of Brownian motion with rank-dependent drift and diffusions connected to the generalised Pitman transform.