Non-homogeneous random walks. Lyapunov function methods for near-critical stochastic systems. (English) Zbl 1376.60005

Menshikov, Mikhail; Popov, Serguei; Wade, Andrew

Cambridge Tracts in Mathematics 209. Cambridge: Cambridge University Press (ISBN 978-1-107-02669-8/hbk; 978-1-139-20846-8/ebook). xviii, 363 p. (2017).

This is another impressive volume in the prestigious ‘Cambridge Tracts in Mathematics’ series. The topic is near-critical stochastic systems based on multidimensional non-homogeneous random walks. These systems are essentially more complex than the classical simple random walk and the big success of the authors is using the modern and powerful semimartingale and Lyapunov functions method.
The authors of this book are well-known for their long standing and well-recognized contributions to this area of research. Besides their own results published over the last two decades, the authors cover all significant achievements up to date.
The books starts with a careful explanation of the models and their complexity, the questions arising, the methods for finding answers, and various possible applications of both models and results.
The material is well structured and presented in seven chapters. Here are the names of the chapters and key words of what is in each chapter: 1. Introduction (simple random walk, recurrence and transience, Lamperti’s problem for \(d\)-dimensional random walk). 2. Semimartingale approach and Markov chains (semimartingale facts, hitting times and their moments). 3. Lamperti’s problem (Markov chains, Lyapunov functions, recurrence, irreducibility, moments and tails of passage times, transience in the critical case, supercritical case). 4. Many-dimensional random walks (elliptic, controlled driftless and centrally biased random walks, range and local time). 5. Heavy tails (directional transience, oscillating random walks). 6. Further applications (random walks and random strings in random environment, stochastic billiards, exclusion and voter models). 7. Markov chains in continuous time (recurrence and transience, moments of passage times, explosion and implosion).
In any section or sub-section, we find precise definitions of the new notions and description of their properties. The results, lemmas and theorems, are clearly formulated and given with compact and complete proofs. A large number of illustrative examples support well the statements and show their relevance to applied areas.
It is remarkable to see detailed ‘Bibliographical notes’ at the end of each chapter. The authors have done a great job by providing valuable information about the historical development of any topic treated in this book. We find extremely interesting facts, stories and references. All this makes the book more than interesting to read and use.
In many places we see texts, in italics, marked by \(\mathbf i\) in a square, giving a good explanation of the meaning of notions, conditions, or statements; e.g., Lyapunov functions, single big jump phenomenon, transformation of near-critical processes, etc.
Some ‘named’ conditions given in the text and frequently used in other parts of the book are collected in a special ‘Glossary’. The book ends with a comprehensive list of 321 references and Index.
Reviewer’s remark. While the font \(\mathbb R\), \(\mathbb N\) is traditional for the sets of real numbers, natural numbers, why to use the same font \(\mathbb P\) and \(\mathbb E\) for probability and expectation? The idea is: use different clothes for different bodies! The ‘bold’ font \(\mathbf P\) and \(\mathbf E\) is good and makes everything easily distinguishable in the text. However the best is the font called ‘text sans serif’: \(\mathsf{P, \, E}\). It is used by many other outstanding authors.
There are all reasons to strongly recommend this book not only to institutional libraries, but also to graduate university students, PhD students and professionals. Researchers in applied areas will benefit a lot from this excellent book.

Reviewer: Jordan M. Stoyanov (Ljubljana)

MSC:

60-02	Research exposition (monographs, survey articles) pertaining to probability theory
60G48	Generalizations of martingales
60G60	Random fields

Keywords:

Lamperti’s problem; \(d\)-dimensional random walk; Markov chains; Lyapunov functions