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.
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.
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.
A modern presentation of the 'Lyapunov function' method applied to near-critical stochastic systems, exemplified by non-homogeneous random walks.
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.
'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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