Menshikov M., Popov S., Wade A. Non-homogeneous Random Walks: Lyapunov Function Methods for Near-Critical Stochastic Systems

Cambridge: Cambridge University Press, 2016. — 383 p.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.NotationRandom Walks
Simple Random Walk
Lamperti’s Problem
General Random Walk
Recurrence and Transience
Angular Asymptotics
Centrally Biased Random Walks
Bibliographical Notes
Semimartingale Approach and Markov Chains
Definitions
An Introductory Example
Fundamental Semimartingale Facts
Displacement and Exit Estimates
Recurrence and Transience Criteria for Markov Chains
Expectations of Hitting Times and Positive Recurrence
Moments of Hitting Times
Growth Bounds on Trajectories
Bibliograpical Notes
Lamperti’s ProblemMarkovian Case
General Case
Lyapunov Functions
Recurrence Classification
Irreducibility and Regeneration
Moments and Tails of Passage Times
Excursion Durations and MAXIMA
Almost-Sure Bounds on Trajectories
Transient Theory in the Critical Case
Nullity and Weak Limits
Supercritical Case
Proofs for the Markovian Case
Bibliographical Notes
Many-Dimensional Random WalksElliptic Random Walks
Controlled Driftless Random Walks
Centrally Biased Random Walks
Range and Local Time of Many-Dimensional Martingales
Bibliographical Notes
Heavy Tails
Chapter Overview
Directional Transience
Oscillating Random Walk
Bibliographical Notes
Further Applications
Random Walk in Random Environment
Random Strings in Random Environment
Stochastic Billiards
Exclusion and Voter Models
Bibliographical Notes
Markov Chains in Continuous Time
Introduction and Notation
Recurrence and Transience
Existence and Non-existence of Moments of Passage Times
Explosion and Implosion
Applications
Bibliographical Notes
Glossary of Named Assumptions