Characterization of stationary distributions of reflected diffusions

Abstract

Given a domain $G$, a reflection vector field $d(\cdot)$ on $\partial G$, the boundary of $G$, and drift and dispersion coefficients $b(\cdot)$ and $\sigma(\cdot)$, let $\mathcal{L}$ be the usual second-order elliptic operator associated with $b(\cdot)$ and $\sigma(\cdot)$. 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 $\pi$ on $\bar{G}$ with $\pi(\partial G)=0$ is a stationary distribution for the corresponding reflected diffusion if and only if

\[\int_{\bar{G}}\mathcal{L}f(x)\pi(dx)\leq0\]

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 $\mathcal{C}^{1}$, and the covariance matrix is nondegenerate, then the corresponding reflected diffusion has an invariant density $p$ of Gibbs form, that is, $p(x)=e^{H(x)}$ for some $\mathcal{C}^{2}$ function $H$. Finally, under a nondegeneracy condition on the diffusion coefficient, a boundary property is established that implies that the condition $\pi(\partial G)=0$ is necessary for $\pi$ to be a stationary distribution. This boundary property is of independent interest.