1 - Introduction

Summary

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.