# Denumerable Markov Chains: Generating Functions, Boundary Theory, Random Walks on Trees

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*Wolfgang Woess*

A publication of the European Mathematical Society

Markov chains are among the basic and most important examples of
random processes. This book is about time-homogeneous Markov chains that evolve
with discrete time steps on a countable state space.

A specific feature is the systematic use, on a relatively
elementary level, of generating functions associated with transition
probabilities for analyzing Markov chains. Basic definitions and facts
include the construction of the trajectory space and are followed by
ample material concerning recurrence and transience, the convergence
and ergodic theorems 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 mathematical models of
population evolution such as birth-and-death chains,
Galton–Watson process and branching Markov chains.

A good part of the second half is devoted to the introduction of
the basic language and elements of the potential theory of transient
Markov chains. Here the construction and properties of the Martin
boundary for describing positive harmonic functions are crucial. In
the long final chapter on nearest neighbor random walks on (typically
infinite) trees the reader can harvest from the seed of methods laid
out so far, in order to 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's degree students to researchers in mathematics,
and persons who want to teach the subject on a medium or advanced
level. Measure theory is not avoided; careful and complete proofs are
provided. A specific characteristic of the book is the rich source of
classroom-tested exercises with solutions.

A publication of the European Mathematical Society (EMS). Distributed within the Americas by the American Mathematical Society.

#### Readership

Graduate students and research mathematicians interested in Markov chains.