A Classification of Diffusion Processes with Boundaries by their Invariant Measures

Abstract

Let $D$ be a connected, compact region in $R^d$. If $d = 1$, then for each nice probability measure $\mu$ on $D$ and diffusion coefficient $a$, there exists a unique drift such that $\mu$ is invariant for the resulting diffusion process with reflection at the boundary. For $d > 1$, there is no uniqueness. For each diffusion matrix $a$, reflection vector $J$, and nice probability measure $\mu$ on $D$, we classify the collection of drifts such that $\mu$ is invariant for the resulting diffusion process. We use the theory of the $I$-function and, in the course of things, answer a question about the $I$-function.