Ergodic Markov Processes and Poisson Equations (Lecture Notes)

These are lecture notes on the subject defined in the title. As such, they do not pretend to be really new, perhaps, except for the Sect. 10 about Poisson equations with potentials; also, the convergence rate shown in (83)–(84) is possibly less known. Yet, the hope of the author is that these notes 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 including (Ethier and Kurtz, Markov Processes: Characterization and Convergence, New Jersey, 2005) [12], (Papanicolaou et al. Statistical Mechanics, Dynamical Systems and the Duke Turbulence Conference, vol. 3. Durham, N.C., 1977) [34], (Pardoux and Veretennikov, Ann. Prob. 31(3), 1166–1192, 2003) [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 and convergence rates in the Ergodic theorem.