Author Description

This book is devoted exclusively to a very special class of random processes, namely to random walk on the lattice points of ordinary Euclidean space. I considered this high degree of specialization worth while, because the theory of such random walks is far more complete than that of any larger class of Markov chains. Random walk occupies such a privileged position primarily because of a delicate interplay between methods from harmonic analysis on one hand, and from potential theory on the other. The relevance of harmonic analysis to random walk of course stems from the invariance of the transition probabilities under translation in the additive group which forms the state space. It is precisely for this reason that, until recently, the subject was dominated by the analysis of characteristic functions (Fourier transforms of the transition probabilities). But if harmonic analysis were the central theme of this book, then the restriction to random walk on the integers (rather than on the reals, or on o'ther Abelian groups) would be quite unforgivable. Indeed it was the need for a self contained elementary exposition of the connection of harmonic analysis with the much more recent developments in potential theory that dictated the simplest possible setting.