Characterization of stationary distributions of reflected diffusions

Given a domain G, a reflection vector field d(·) on ∂ G, 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 mild assumptions on the coefficients and reflection vector field, it is shown that when the associated submartingale problem is well posed, a probability measure π on Ḡ with π(∂ G)=0 is a stationary distribution for the corresponding reflected diffusion if and only if \[∫ḠLf(x)π(dx)≤0\] for every f in a certain class of test functions. The assumptions are verified for a large class of obliquely reflected diffusions in piecewise smooth domains, including those that are not semimartingales. In addition, it is shown that any nonnegative solution to a certain adjoint partial differential equation with boundary conditions is an invariant density for the reflected diffusion. As a corollary, for bounded smooth domains and a class of polyhedral domains that satisfy a skew-symmetry condition, it is shown that if a certain skew-transform of the drift is conservative and of class C¹, and the covariance matrix is nondegenerate, then the corresponding reflected diffusion has an invariant density p of Gibbs form, that is, p(x)=eH(x) for some C² function H. Finally, under a nondegeneracy condition on the diffusion coefficient, a boundary property is established that implies that the condition π(∂ G)=0 is necessary for π to be a stationary distribution. This boundary property is of independent interest.