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

TL;DR

This paper investigates Poisson equations with a potential term for ergodic generators in the whole space, aiming to extend PDE theory and applications in stochastic homogenization and averaging.

Contribution

It introduces a Poisson equation with a potential for ergodic generators, expanding the theoretical framework and potential applications in homogenization and stochastic averaging.

Findings

Provides a new formulation of Poisson equations with potentials

Lays groundwork for applications in homogenization and averaging

Extends previous ergodic generator analysis

Abstract

In earlier papers Poisson equation in the whole space was studied for so called ergodic generators $L$ corresponding to homogeneous Markov diffusions ($X_t, \, t\ge 0$) in $\mathbb R^d$. Solving this equation is one of the main tools for diffusion approximation in the theory of stochastic averaging and homogenisation. Here a similar equation with a potential is considered, firstly because it is natural for PDEs, and secondly with a hope that it may be also useful for some extensions related to homogenization and averaging.