Non-homogeneous Random Walks: Lyapunov Function Methods for Near-Critical Stochastic Systems

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Mikhail Menshikov (University of Durham) Serguei Popov (Universidade Estadual de Campinas, Brazil)

Non-homogeneous Random Walks

Lyapunov Function Methods for Near-Critical Stochastic Systems

Mikhail Menshikov (University of Durham), Serguei Popov (Universidade Estadual de Campinas, Brazil), Andrew Wade (University of Durham)

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English

Cambridge University Press
22 December 2016

Groups & group theory; Probability & statistics; Stochastics

Series: Cambridge Tracts in Mathematics

Stochastic systems provide powerful abstract models for a variety of important real-life applications: for example, power supply, traffic flow, data transmission. They (and the real systems they model) are often subject to phase transitions, behaving in one way when a parameter is below a certain critical value, then switching behaviour as soon as that critical value is reached. In a real system, we do not necessarily have control over all the parameter values, so it is important to know how to find critical points and to understand system behaviour near these points. This book is a modern presentation of the 'semimartingale' or 'Lyapunov function' method applied to near-critical stochastic systems, exemplified by non-homogeneous random walks. Applications treat near-critical stochastic systems and range across modern probability theory from stochastic billiards models to interacting particle systems. Spatially non-homogeneous random walks are explored in depth, as they provide prototypical near-critical systems.

By: Mikhail Menshikov (University of Durham), Serguei Popov (Universidade Estadual de Campinas, Brazil), Andrew Wade (University of Durham)
Imprint: Cambridge University Press
Country of Publication: United Kingdom
Volume: 209
Dimensions: Height: 237mm, Width: 160mm, Spine: 30mm
Weight: 730g
ISBN: 9781107026698
ISBN 10: 1107026695
Series: Cambridge Tracts in Mathematics
Pages: 382
Publication Date: 22 December 2016
Audience: Professional and scholarly , Undergraduate
Format: Hardback
Publisher's Status: Active

Mikhail Menshikov is Professor in the Department of Mathematical Sciences at the University of Durham. His research interests include percolation theory, where Menshikov's theorem is a cornerstone of the subject. He has published extensively on the Lyapunov function method and its application, for example to queueing theory. Serguei Popov is Professor in the Department of Statistics, Institute of Mathematics, Statistics and Scientific Computation, Universidad Estadual de Campinas, Brazil. His research interests include several areas of probability theory, besides Markov chains, including percolation, stochastic billiards, random interlacements, branching processes, and queueing models. Andrew Wade is Senior Lecturer in the Department of Mathematical Sciences at the University of Durham. His research interests include, in addition to random walks, interacting particle systems, geometrical probability, and random spatial structures.

Reviews for Non-homogeneous Random Walks: Lyapunov Function Methods for Near-Critical Stochastic Systems

'This is another impressive volume in the prestigious `Cambridge Tracts in Mathematics' series ... 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 ... 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.' Jordan M. Stoyanov, Zentralblatt MATH