Ergodic Markov processes and Poisson equations (lecture notes)

TL;DR

This lecture note introduces ergodic Markov processes and Poisson equations, emphasizing their role in diffusion approximation and stochastic averaging, with a focus on bridging discrete and continuous operator contexts.

Contribution

It provides an accessible overview of Poisson equations with potentials and discusses their significance in diffusion approximation and stochastic averaging.

Findings

Highlights the importance of Poisson equations in stochastic analysis.

Introduces coupling method for ergodic Markov processes.

Serves as a bridge to differential operator-based Poisson equations.

Abstract

These are lecture notes on the subject defined in the title. As such, they do not pretend to be really new, probably except for the only section about Poisson equations with potentials. Yet, the hope of the author is that they may serve as a bridge to the important area of Poisson equations "in the whole space" and with a parameter, the latter theme not being presented here. Why this area is so important was explained in many papers and books (see the references [12, 34, 35]): it provides one of the main tools in diffusion approximation in the area stochastic averaging. Hence, the aim of these lectures is to prepare the reader to "real" Poisson equations -- i.e., for differential operators instead of difference operators -- and, indeed, to diffusion approximation. Among other presented topics we mention coupling method.