Look Inside Non-homogeneous Random Walks

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

Part of Cambridge Tracts in Mathematics

Authors:
Mikhail Menshikov, University of Durham
Serguei Popov, Universidade Estadual de Campinas, Brazil
Andrew Wade, University of Durham

Date Published: December 2016
availability: Available
format: Hardback
isbn: 9781107026698

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Description

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.
- Provides a deep understanding of the Lyapunov function method applied to a prototypical near-critical system
- Foundational theory and applications are illustrated by examples and with remarks on intuition
- Serves as a reference for experts as well as an instructional course for younger researchers
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Reviews & endorsements
'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
'This book gives a comprehensive account of the study of random walks with spatially non-homogeneous transition kernels. The main theme is to study recurrence versus transience and moments of passage times, as well as path asymptotics, by constructing suitable Lyapunov functions, which deﬁne semi-martingales when composed with the random walk. Of special interest are the Lamperti processes, which are stochastic processes on [0, ∞) with drift vanishing asymptotically on the order of 1/x as the space variable x tends to inﬁnity. … Each chapter ends with detailed bibliographical notes.' Rongfeng Sun, Mathematical Reviews
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Product details
- Date Published: December 2016
- format: Hardback
- isbn: 9781107026698
- length: 382 pages
- dimensions: 237 x 160 x 30 mm
- weight: 0.73kg
- contains: 20 b/w illus.
- availability: Available
Table of Contents
1. Introduction
2. Semimartingale approach and Markov chains
3. Lamperti's problem
4. Many-dimensional random walks
5. Heavy tails
6. Further applications
7. Markov chains in continuous time
Glossary of named assumptions
Bibliography
Index.
Look Inside
Authors
Mikhail Menshikov, University of Durham
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, Universidade Estadual de Campinas, Brazil
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, University of Durham
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.