Non-homogeneous Random Walks: Lyapunov Function Methods for Near-Critical Stochastic Systems (Cambridge Tracts in Mathematics, Band 209)

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. Contents Preface Notation 1 Introduction 1.1 Random Walks 1.2 Simple Random Walk 1.3 Lamperti’s Problem 1.4 General Random Walk 1.5 Recurrence and Transience 1.6 Angular Asymptotics 1.7 Centrally Biased Random Walks Bibliographical Notes 2 Semimartingale Approach and Markov Chains 2.1 Definitions 2.2 An Introductory Example 2.3 Fundamental Semimartingale Facts 2.4 Displacement and Exit Estimates 2.5 Recurrence and Transience Criteria for Markov Chains 2.6 Expectations of Hitting Times and Positive Recurrence 2.7 Moments of Hitting Times 2.8 Growth Bounds on Trajectories Bibliograpical Notes 3 Lamperti’s Problem 3.1 Introduction 3.2 Markovian Case 3.3 General Case 3.4 Lyapunov Functions 3.5 Recurrence Classification 3.6 Irreducibility and Regeneration 3.7 Moments and Tails of Passage Times 3.8 Excursion Durations and Maxima 3.9 Almost-Sure Bounds on Trajectories 3.10 Transient Theory in the Critical Case 3.11 Nullity and Weak Limits 3.12 Supercritical Case 3.13 Proofs for the Markovian Case Bibliographical Notes 4 Many-Dimensional Random Walks 4.1 Introduction 4.2 Elliptic Random Walks 4.3 Controlled Driftless Random Walks 4.4 Centrally Biased Random Walks 4.5 Range and Local Time of Many-Dimensional Martingales Bibliographical Notes 5 Heavy Tails 5.1 Chapter Overview 5.2 Directional Transience 5.3 Oscillating Random Walk Bibliographical Notes 6 Further Applications 6.1 Random Walk in Random Environment 6.2 Random Strings in Random Environment 6.3 Stochastic Billiards 6.4 Exclusion and Voter Models Bibliographical Notes 7 Markov Chains in Continuous Time 7.1 Introduction and Notation 7.2 Recurrence and Transience 7.3 Existence and Non-existence of Moments of Passage Times 7.4 Explosion and Implosion 7.5 Applications Bibliographical Notes Glossary of Named Assumptions References Index