Characterization of stationary distributions of reflected diffusions

Given a domain G, a reflection vector field d(.) on the boundary of G, and drift and dispersion coefficients b(.) and σ(.), let L be the usual second-order elliptic operator associated with b(.) and σ(.). Under suitable assumptions that, in particular, ensure that the associated submartingale problem is well posed, it is shown that a probability measure $π$ on \bar{G} is a stationary distribution for the corresponding reflected diffusion if and only if $π(\partial G) = 0$ and $\int_{\bar{G}} L f (x) π(dx) \leq 0$ 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.

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