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

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.