Customer Reviews

A Guide to First-Passage Processes

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An engaging survey of first passage time processes, highlighting interesting consequences, such as the nonmonotonic behavior of stochastic resonance and the application of first-passage methods to efficient simulation of diffusion-limited aggregation. The book assumes familiarity with diffusion, generating functions, Laplace transforms and asymptotics. Readers without that background will struggle with the derivations, but can appreciate the results. For example, the connection between discrete random walks and continuous diffusion are stated without much motivation in section 1.3.3. The provided references are good for mathematically oriented readers. Others could benefit from a more elementary presentation, e.g., Random Walks in Biology.

The book occasionally mentions significant extensions without even briefly describing their consequences. This is unfortunate since extensions such as the Orstein-Uhlenbeck process apply widely, e.g., to neuron models mentioned in the text as well as to finance (where an extended discussion would connect nicely with the stock market example described on the first page of the book).

The book is well-organized, but the index lacks common alternate names for processes mentioned in the text, such as "integrated Brownian motion", "Orstein-Uhlenbeck process" and "partially absorbing boundary condition".

The book would benefit from a summary of results to highlight in one place the relationships among the applications. Surveying open problems would make an interesting conclusion to this summary. The book could also use a table of notation, particularly since transforms are indicated by the name of arguments: e.g., for a discrete random walk, P(k,t) denotes the Fourier transform of the occupation probability P(n,t) to be at site n at time t, instead of the occupation probability at a different location k. This notation for transforms is handled consistently, but may confuse casual readers.

The more complicated derivations (not shown in the text) rely on Mathematica. It would be helpful to make the Mathematica files available on the book's web site so interested readers could see the details. The 2007 paperback edition has several pages of errata (which, surprisingly, are more recent than the list on the book's web site). While this list is helpful, it requires constantly checking while reading. I was surprised that the paperback edition did not correct the text.

Professor Redner gives us fair warning on page 3: "Note that we are using the terms random walk and diffusing particle loosely and interchangeably. Although these two processes are very different microscopically, their long-time properties--including first-passage characteristics--are essentially the same." As promised, he then proceeds to blur the difference between discrete time and continuous time and the difference between discrete space and continuous space. He treats probability mass functions and probability density functions, which he calls particle concentrations, as equivalent. He applies generating functions, which are defined for discrete time, to functions defined on continuous time. Section 1.5 is particularly bad in this regard. If you're not that interested in being accurate, willing to endure a headache picking the nuggets from the tailings or seeking to confirm your belief that mathematicians and scientists put their best writing in peer-reviewed journals, read this book. Otherwise, skip it.

I took advanced statistical physics from Professor Redner (the author) and highly recommend this book. He is without a doubt one of the best teachers I have ever had.

... formulated both discretely and continuously. Sidney Redner provides us with a stimulating and masterful treatment of first passage times from the standpoint of traditional statistical physics. This is both the strength and weakness of his attractive book. What's lacking is a systematic formulation with simple examples continuous in (x,t) that the reader can solve for herself. The formulation of the hitting probability for the simplest diffusion problem (the Wiener process) as a problem in electrostatics (pg. 24) is nice but woefully incomplete and inadequate, and is related to Durrett's ('Brownian Motion and Martingales in Analysis', 1984) attempts to formulate potential problems via Brownian motion averages using stopping times. What's missing in Redner's electrostatics approach can be found in Stratonovich's "Topics in the Theory of Random Noise" Vol. I: the use of Kolmogorov's backward time diffusion pde to calculate the average stopping time for general time translational invariant diffusion problems with variable drift and diffusio coefficients. And see Steele's "Stochastic Calculus" for a formulation of hitting times (stopping times) using Martingales formulated via Ito calculus and stochastic differential equations. Steele provides many interesting and instructive examples, including a simple calculation ofthe average stopping time for Wiener processes with and without drift (the Durrett-Steele Ito approach is much shorter than Stratonovich's Fokker-Planck method, and both approaches are instructive). Actually, Durrett (1984) shows assigns the most general method as 2 exerecises on pg. 255: to calculate the average hitting time by constructing a martingale that leads to solving the same backward tiome pde as Stratonovich, but for the general case of (x,t) dependent drift and diffusion coefficients!