This book gives an introduction to discrete-time Markov chains which evolve on a separable metric space. The focus is on the ergodic properties of such chains, i.e., on their long-term statistical behaviour. Among the main topics are existence and uniqueness of invariant probability measures, irreducibility, recurrence, regularizing properties for Markov kernels, and convergence to equilibrium. These concepts are investigated with tools such as Lyapunov functions, petite and small sets, Doeblin and accessible points, coupling, as well as key notions from classical ergodic theory. The theory is illustrated through several recurring classes of examples, e.g., random contractions, randomly switched vector fields, and stochastic differential equations, the latter providing a bridge to continuous-time Markov processes. The book can serve as the core for a semester- or year-long graduate course in probability theory with an emphasis onMarkov chains or random dynamics. Some of the material is also well suited for an ergodic theory course. Readers should have taken an introductory course on probability theory, based on measure theory. While there is a chapter devoted to chains on a countable state space, a certain familiarity with Markov chains on a finite state space is also recommended.

Autorentext

Michel Benaïm is a full professor and the head of the probability group at the University of Neuchâtel. He has taught at the universities of Toulouse, Cergy-Pontoise, Ecole Normale Supérieure de Cachan (now Paris-Saclay) and Ecole Polytechnique. Together with Nicole El Karoui, he is the author of the textbook Promenade Aléatoire. He has worked extensively on problems at the interface of probability theory and dynamical systems. He is a member of the editorial boards of Journal of Dynamics and Games, the Springer collection Mathématiques et Applications, and Stochastic Processes and their Applications. Tobias Hurth received his Ph.D. in mathematics from the Georgia Institute of Technology in 2014. He has since held postdoctoral positions at the University of Toronto, the Ecole Polytechnique Fédérale de Lausanne, and the University of Neuchâtel. His research interests include stochastic processes, random dynamics, and mathematical physics.