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This is the revised and augmented edition of a now classic book which is an introduction to sub-Markovian kernels on general measurable spaces and their associated homogeneous Markov chains. The first part, an expository text on the foundations of the subject, is intended for post-graduate students. A study of potential theory, the basic classification of chains according to their asymptotic behaviour and the celebrated Chacon-Ornstein theorem are examined in detail.
The second part of the book is at a more advanced level and includes a treatment of random walks on general locally compact abelian groups. Further chapters develop renewal theory, an introduction to Martin boundary and the study of chains recurrent in the Harris sense. Finally, the last chapter deals with the construction of chains starting from a kernel satisfying some kind of maximum principle.
List of contents
Transition Probabilities. Markov Chains. Potential Theory. Transcience and Recurrence. Pointwise Ergodic Theory. Transient Random Walks. Ergodic Theory of Harris Chains. Martin Boundary. Potential Theory for Harris Chains. Recurrent Random Walks. Construction of Markov Chains and Resolvents.
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"...carefully organized, the proofs are well thought, thorough and elegant. Many problems supplement the material in the text... The first few chapters provide a good introduction to Markov chains for the novice and most experts will find something new in the latter chapters." --Bulletin of the American Mathematical Society
"...the book provides a good summary of the development of the subject in the past twenty years. Its careful organization, so convenient for the reader, has already recommended it as a text-book for various courses and seminars." --Mathematical Reviews
