EMS Textbooks in Mathematics

Wolfgang Woess (Graz University of Technology, Austria)

Denumerable Markov ChainsGenerating Functions, Boundary Theory, Random Walks on Trees

Markov chains are the first and most important examples of random processes.This book is about time-homogeneous Markov chains that evolve with discrete timesteps on a countable state space. Measure theory is not avoided, careful andcomplete proofs are provided.
A specific feature is the systematic use, on a relatively elementary level, of generatingfunctions associated with transition probabilities for analyzing Markov chains. Basicdefinitions and facts include the construction of the trajectory space and are followedby ample material concerning recurrence and transience, the convergence and ergodictheorems for positive recurrent chains. There is a side-trip to the Perron–Frobenius theorem.Special attention is given to reversible Markov chains and to basic mathematicalmodels of “population evolution” such as birth-and-death chains, Galton–Watsonprocess and branching Markov chains.
A good part of the second half is devoted to the introduction of the basic languageand elements of the potential theory of transient Markov chains. Here the constructionand properties of the Martin boundary for describing positive harmonic functionsare crucial. In the long final chapter on nearest neighbour random walks on (typicallyinfinite) trees the reader can harvest from the seed of methods laid out so far, in orderto obtain a rather detailed understanding of a specific, broad class of Markov chains.
The level varies from basic to more advanced, addressing an audience from master’sdegree students to researchers in mathematics, and persons who want to teach thesubject on a medium or advanced level. A specific characteristic of the book is the richsource of classroom-tested exercises with solutions.