Dedicated to N. V. Krylov on his sixtieth birthdayA Poisson equation in d for the elliptic operator corresponding to an ergodic diffusion process is considered. Existence and uniqueness of its solution in Sobolev classes of functions is established along with the bounds for its growth. This result is used to study a diffusion approximation for two-scaled diffusion processes using the method of corrector; the solution of a Poisson equation serves as a corrector.

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We study the Poisson equation Lu+f=0 in R^d, where L is the infinitesimal generator of a diffusion process. In this paper, we allow the second-order part of the generator L to be degenerate, provided a local condition of Doeblin type is satisfied, so that, if we also assume a condition on the drift which implies recurrence, the diffusion process is ergodic. The equation is understood in a weak sense. Our results are then applied to diffusion approximation.Comment: Published at http://dx.doi.org/10.1214/009117905000000062 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

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New weak and strong existence and weak and strong uniqueness results for multi-dimensional stochastic McKean-Vlasov equation are established under relaxed regularity conditions. Weak existence is a variation of Krylov's weak existence for Itô's SDEs under the nondegeneracy of diffusion and no more than a linear growth in the state variable; this part is designed to fill in the existing gap, as earlier such results for McKean-Vlasov equations were not written. Weak and strong uniqueness is established under the restricted assumption of diffusion depending only on time and the state variable, yet without any regularity of the drift with respect to the state variable and also under a linear growth condition on this drift; this part is based on the analysis of the total variation metric.