A guide to first-passage processes.

*(English)*Zbl 0980.60006
Cambridge: Cambridge University Press. ix, 312 p. (2001).

The book deals with the first-passage properties of random walks and diffusion. The key point is the first-passage probability that a diffusing particle or a random walk first reaches a specific site (or set of sites) at a specified time. There are many applications of such first-passage phenomena of stochastic processes in electrostatics, chemistry, kinetics, finance (e.g., financial options) and many other disciplines. The author’s goal is to help those with modest background of the first-passage theory to learn essential results quickly (e.g., graduate students and researchers in physics, theoretical biology, electrical and chemical engineering, operation research, finance).

Chapter 1 (First-passage fundamentals) provides the fundamental background and connections (e.g. to electrostatics). Chapter 2 (First-passage in an interval) discusses first passage in a one-dimensional interval. Chapter 3 (Semi-infinite systems) treats first passage in a semi-infinite interval. Chapter 4 (Illustrations of first-passage in simple geometries) is devoted to various practical illustrations (neuron dynamics, realization of self-organized criticality, kinetics of spin systems, resonant escape from fluctuating and inhomogeneous media, survival of a diffusing particle in an expanding cage and near a moving cliff). Chapter 5 (Fractal and nonfractal networks) turns to first passage on branched, self-similar structures. Chapter 6 (Systems with spherical symmetry) treats spherically symmetric geometries. First passage in wedge and conical domains are presented in Chapter 7 (Wedge domains) and kinetics of one-dimensional diffusion-controlled reactions in Chapter 8 (Applications to simple reactions).

Chapter 1 (First-passage fundamentals) provides the fundamental background and connections (e.g. to electrostatics). Chapter 2 (First-passage in an interval) discusses first passage in a one-dimensional interval. Chapter 3 (Semi-infinite systems) treats first passage in a semi-infinite interval. Chapter 4 (Illustrations of first-passage in simple geometries) is devoted to various practical illustrations (neuron dynamics, realization of self-organized criticality, kinetics of spin systems, resonant escape from fluctuating and inhomogeneous media, survival of a diffusing particle in an expanding cage and near a moving cliff). Chapter 5 (Fractal and nonfractal networks) turns to first passage on branched, self-similar structures. Chapter 6 (Systems with spherical symmetry) treats spherically symmetric geometries. First passage in wedge and conical domains are presented in Chapter 7 (Wedge domains) and kinetics of one-dimensional diffusion-controlled reactions in Chapter 8 (Applications to simple reactions).

Reviewer: T.Cipra (Praha)