On Poisson equation and diffusion approximation 2

Abstract

Three different results are established which turn out to be closely connected so that the first one implies the second one which in turn implies the third one. The first one states the smoothness of an invariant diffusion density with respect to a parameter. The second establishes a similar smoothness of the solution of the Poisson equation in $\mathbb{R}^d$. The third one states a diffusion approximation result, or in other words an averaging of singularly perturbed diffusion for "fully coupled SDE systems'' or "SDE systems with complete dependence.''