A dual skew symmetry for transient reflected Brownian motion in an orthant

TL;DR

This paper studies a transient reflected Brownian motion in a multidimensional orthant, providing conditions for absorption probability to have an exponential product form, linked to a dual skew symmetry related to the reflection matrix.

Contribution

It introduces the concept of dual skew symmetry for transient reflected Brownian motion and characterizes when the absorption probability has an exponential product form.

Findings

Absorption probability admits an exponential product form if and only if the reflection matrix determinant is zero.

The dual skew symmetry condition generalizes the classical skew symmetry for stationary distributions.

The PDE for absorption probability is dual to that for stationary distribution in the recurrent case.

Abstract

We introduce a transient reflected Brownian motion in a multidimensional orthant, which is either absorbed at the apex of the cone or escapes to infinity. We address the question of computing the absorption probability, as a function of the starting point of the process. We provide a necessary and sufficient condition for the absorption probability to admit an exponential product form, namely, that the determinant of the reflection matrix is zero. We call this condition a dual skew symmetry. It recalls the famous skew symmetry introduced by Harrison, which characterizes the exponential stationary distributions in the recurrent case. The duality comes from that the partial differential equation satisfied by the absorption probability is dual to the one associated with the stationary distribution in the recurrent case.