Pardoux and Veretennikov [2001, Ann. Probab. 29, 1061-1085] establish existence (in the Sobolev sense) as well as uniqueness of solutions of a Poisson equation for the differential operator of an ergodic diffusion in Euclidean space, for which the covariance term is strictly non-degenerate and uniformly bounded. Our goal is to establish a similar solvability result, but for the complementary case of singular diffusions, in which the covariance is not necessarily of full-rank. In return for abandoning strict positive- definiteness of the covariance, we postulate second-order smoothness of the coefficients of the diffusion, and, to secure ergodicity, we postulate a "stability" condition on the eigenvalues of the symmetrized Jacobian matrix of the drift. We establish existence of solutions of the Poisson equation in the classical sense, and use this to characterize limits in a multiple time-scales problem arising from perturbation of a stochastic differential equation by a rescaled singular diffusion. This is motivated by a stochastic averaging principle of Liptser and Stoyanov.
Groupe d’études et de recherche en analyse des décisions