Markov chains on metric spaces. A short course (Q2170473)

The Markov chains play a fundamental role in probability theory and its many applications already for more than a century. Available today is a beautiful mathematical theory, and the present book is one evidence more. The book is based on the experience of the authors of teaching graduate university courses. Not less important is their active research and publications in Markov processes and models. Let us list the names of the chapters and in brackets just the name of one section: 1. Markov chains (Markov and strong Markov property). 2. Countable Markov chains (Recurrence and Lyapunov functions). 3. Random dynamical systems (representation of Markov chains by RDS). 4. Invariant and ergodic probability measures (classical results from ergodic theory: Poincaré, Birkhoff, and ergodic decomposition). 5. Irreducibility (the asymptotic strong Feller property). 6. Petite sets and Doeblin points (piecewise deterministic Markov processes). 7. Harris and positive recurrence (recurrence criteria and Lyapunov functions). 8. Harris ergodic theorem (convergence in Wasserstein distance). Appendix: Monotone class and martingales. Bibliography, List of symbols, Index. The book is written in a rigorous style. Any new notion is defined and its properties described in statements, propositions and theorems, followed by compact proofs. There is a large number of very useful exercises, in some cases with short hints. In order to be successful the reader needs a strong mathematical background. As a level, the material in this book is suitable for master programs at good universities. Solving the exercises would take serious efforts, however this will guarantee a great knowledge in Markov chains in metric spaces and their ergodicity. Many university teachers giving courses in Markov chains, random dynamics or ergodic theory, may find useful to combine their own favourite sources with this new and challenging book. In all cases the book can be strongly recommended for further study in this area.