Uniqueness properties of diffusion processes

Abstract

We review recent results on the uniqueness of solutions of the diffusion equation \[ \partial \psi_{t} / \partial t + H \psi_{t} = 0 \] where $H$ is a strictly elliptic, symmetric, second-order operator on an open subset $\Omega$ of $\mathbf{R}^d$. In particular we discuss $L_1$-uniqueness, the existence of a unique continuous solution on $L_1(\Omega)$, and Markov uniqueness, the existence of a unique submarkovian solution on the spaces $L_p(\Omega)$. We give various criteria for uniqueness in terms of capacity estimates and the Riemannian geometry associated with $H$.