A GUIDE TO FIRST-PASSAGE PROCESSES,
S. Redner, Cambridge University Press, 2001; pp: 307, ISBN 0-521-65248-0, (hc), Price: $80.00.

Consider a Brownian particle undergoing a one-dimensional random walk within a domain between x=0 and x=L, the boundaries being such that the particle will be reflected at x=0 and absorbed at x=L. If the particle starts from some position within the domain when will it hit the absorbing boundary for the first time? This first-passage problem in stochastic processes, in various spatial dimensions and with a variety of geometries and boundary conditions, is of longstanding interest and has ramifications and applications in diverse areas. An early consideration of the first-passage problem is due to Schroedinger (whose work is not mentioned in this book), is published in Physik Zeits (1915), and is discussed in Section 164 of Kinetic Theory of Gases by E.H.Kennard (McGraw-Hill, 1938). The interested reader may like to see how this early discussion compares with a contemporary approach.

In the preface the author states that he has attempted “to give a unified presentation of first-passage processes and illustrate some of the beautiful and fundamental consequences”. The three beginning chapters introduce the first-passage concepts and calculations including the interesting formal connection between first-passage and electrostatics, through a consideration of both discrete and continuous one-dimensional random walk models in finite and semi-infinite intervals and with various boundary conditions. A newcomer to the first-passage problem may like to combine a perusal of these chapters and some introductory textbooks cited by the author in order to have a better appreciation of the contents of these chapters. The first-passage properties in simple geometries are illustrated with some examples of real systems in Chapter 4. The next three chapters move to higher dimensions and consider fractal and non-fractal networks, spherical geometries and wedge domains. Chapter 7, on two-dimensional systems such as wedge domains, carries the connection between first-passage properties and electrostatics further and discusses the powerful conformal transformation methods to deal with these problems. In the context of fractal geometries a curious theoretical development in the last few years consists of the so-called fractional diffusion and Fokker-Planck equations. These are neither utilized nor commented on by the author. Chapter 8, the final chapter, considers applications to diffusion-controlled reactions in which the author and co-workers have been actively involved. An interesting inovation, perhaps too recent to be included in this book, is the problem of persistence, which has a connection to first-passage processes.

The reference list is extensive, although two useful references have not been cited: a review article on first-passage and zero-crossings by I.F. Blake and W.C. Lindsey [Trans. IEEE. Information Theory, Volume 19, pp 295(1973)], and a book by L.M. Ricciardi on “Diffusion Processes and Related Topics in Biology” (Springer-Verlag, 1977) which has a useful chapter on the first-passage problem.

This specialized monograph whose emphasis is mostly on theory and modelling centred on a single physical quantity will not be suitable as a graduate text. But this scholarly book will be of interest to the researchers already working in, and others wishing to enter, the field of stochastic processes. A paperback edition would be desirable.

Amal K. Das
Dalhousie University