Characterization of stationary distributions of reflected diffusions

TL;DR

This paper characterizes stationary distributions of reflected diffusions in multi-dimensional domains, establishing conditions under which a probability measure is stationary, and shows these conditions hold for a broad class of such diffusions.

Contribution

It provides a necessary and sufficient condition for stationary distributions of reflected diffusions, extending to complex multi-dimensional domains with oblique reflection.

Findings

Stationary distributions satisfy specific integral conditions.

Conditions are applicable to a wide class of reflected diffusions.

The framework ensures well-posedness of the submartingale problem.

Abstract

Given a domain G, a reflection vector field d(.) on the boundary of G, and drift and dispersion coefficients b(.) and \sigma(.), let L be the usual second-order elliptic operator associated with b(.) and \sigma(.). Under suitable assumptions that, in particular, ensure that the associated submartingale problem is well posed, it is shown that a probability measure $\pi$ on \bar{G} is a stationary distribution for the corresponding reflected diffusion if and only if $\pi (\partial G) = 0$ and $\int_{\bar{G}} L f (x) \pi (dx) \leq 0$ for every f in a certain class of test functions. Moreover, the assumptions are shown to be satisfied by a large class of reflected diffusions in piecewise smooth multi-dimensional domains with possibly oblique reflection.