Application of the dual-process method to the study of a certain singular diffusion

Abstract

This paper should be regarded as a sequel to a paper by Holley, Stroock and the author. Its primary purpose is to provide further illustration of the application of the dual-process method. The main result is that if $d \geqslant 2$ and $\varphi$ is the characteristic function of an aperiodic random walk on ${{\mathbf {Z}}^d}$, then there is precisely one Feller semigroup on the d-dimensional torus with generator extending $A = \{ 1 - \varphi (\theta )\} \Delta$. A necessary and sufficient condition for the associated Feller process to leave the singular point 0 is determined. This condition provides a criterion for uniqueness in law of a stochastic differential equation which is naturally associated with the process.