On Poisson equations with a potential in the whole space for ergodic'' generators

In earlier works Poisson equation in the whole space was studied for so-called ergodic generators L corresponding to homogeneous Markov diffusions (Xt, t⩾0) in Rd. Solving this equation is one of the main tools for diffusion approximation in the theory of stochastic averaging and homogenization. Here a similar equation with a potential is considered, first because it is natural for PDEs, and second with a hope that it may also be useful for some extensions related to homogenization and averaging.