RandomWalks

Random walks are fundamental models in probability theory that exhibit deep mathematical properties and enjoy broad application across the sciences and beyond. Generally speaking, a random walk is a stochastic process modelling the random motion of a particle (or random walker) in space. The particle's trajectory is described by a series of random increments or jumps at discrete instants in time. Central questions for these models involve the long-time asymptotic behaviour of the walker.

Random walks have a rich history involving several disciplines. Classical one-dimensional random walks were first studied several hundred years ago as models for games of chance, such as the so-called gambler's ruin problem. Similar reasoning led to random walk models of stock prices described by Jules Regnault in his 1863 book [265] and Louis Bachelier in his 1900 thesis [14]. Many-dimensional random walks were first studied at around the same time, arising from the work of pioneers of science in diverse applications such as acoustics (Lord Rayleigh's theory of sound developed from about 1880 [264]), biology (Karl Pearson's 1906 [254] theory of random migration of species), and statistical physics (Einstein's theory of Brownian motion developed during 1905–8 [86]). The mathematical importance of the random walk problem became clear after Pόlya's work in the 1920s, and over the last 60 years or so there have emerged beautiful connections linking random walk theory and other influential areas of mathematics, such as harmonic analysis, potential theory, combinatorics, and spectral theory. Random walk models have continued to find new and important applications in many highly active domains of modern science: see for example the wide range of articles in [287]. Specific recent developments include modelling of microbe locomotion in microbiology [23, 245], polymer conformation in molecular chemistry [15, 202], and financial systems in economics.

Spatially homogeneous random walks are the subject of a substantial literature, including [139, 195, 269, 293]. In many modelling applications, the classical assumption of spatial homogeneity is not realistic: the behaviour of the random walker may depend on the present location in space.

What Is This Book About?

This book has two main goals:

1. • to give an up-to-date exposition of the ‘semimartingale’ or ‘Lyapunov function’ approach to the analysis of stochastic processes;

2. • to present applications of the methodology to fundamental models (classical and modern) in probability theory and related fields.

Our expository bridge between these dual aims, between methods and models, is the d-dimensional non-homogeneous random walk, which as a model is simple to describe, closely resembling the classical homogeneous random walk, but which displays many interesting and subtle phenomena alien to the classical model. Non-homogeneous random walks cannot be studied by the techniques generally used for homogeneous random walks: new methods (and, just as importantly, new intuitions) are required.

Semimartingale and Lyapunov function ideas lead to a unified and powerful methodology in this context. As well as non-homogeneous random walks, we present applications of the methods to several other models from modern probability theory; while any of the models that we discuss can be studied by several probabilistic techniques, we believe that only the Lyapunov function method has something to say about all of them.

We emphasize that semimartingale methods are ‘robust’ in the sense that the underlying stochastic process need not satisfy simplifying assumptions such as the Markov property, reversibility, or time homogeneity, for instance, and the state space of the process need not be countable. In such a general setting, the semimartingale approach has few rivals. In particular, the methods presented work for non-reversible Markov chains. A general feeling is that, if a Markov chain is reversible, then things can be done in many possible ways: there are methods from electrical networks, spectral calculations, harmonic analysis, etc. On the other hand, the non-reversible case is usually much harder. Similarly, the Markovian setting is not essential to the methods. In the semimartingale approach, the Markov property is a side issue and non-Markovian processes can be treated equally well.

Chapter Overview

So far, the processes that we have considered in this book have typically had increments whose first (and second) moments have been uniformly bounded. The focus of this chapter is on the case where first (or second) moments do not exist, i.e., the increments are heavy tailed. We consider Markov processes on R, and study questions of asymptotic behaviour, such as recurrence or transience.

This chapter deals with two different types of process. First, in Section 5.2, we study processes whose increments have, in some sense, heavier tails in one direction than the other.We investigate sufficient conditions for transience in the direction of the heavier tail, and quantify the transience by deriving results on the rate of escape and on moments of first passage and last exit times.

Second, in Section 5.3, we study processes whose increments are of two different types, depending on whether the current position is to the left or to the right of the origin. Such processes are known as oscillating random walks, and their study (via classical methods) goes back to Kemperman [157]. We consider a version of the model in which the increment distribution is signed, i.e., only jumps towards the origin are allowed. For these models, we study recurrence and transience.

In keeping with the theme of this book, the methods of this chapter are based on the semimartingale ideas of Chapter 2: for appropriate choices of Lyapunov function, we verify Foster–Lyapunov-style drift conditions. Verification of drift conditions usually entails some Taylor's formula expansions (as in Chapter 3) as well as some careful truncation ideas to deal with the heavy tails. The resulting proofs are relatively short, and based on some intuitively appealing ideas.

Directional Transience

Introduction

In this section we assume the following.