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A Guide to First-Passage Processes [Hardcover]

Sidney Redner
3.5 out of 5 stars  See all reviews (4 customer reviews)

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Book Description

August 6, 2001 0521652480 978-0521652483
First-passage properties underlie a wide range of stochastic processes, such as diffusion-limited growth, neuron firing, and the triggering of stock options. This book provides a unified presentation of first-passage processes, which highlights its interrelations with electrostatics and the resulting powerful consequences. The author begins with a modern presentation of fundamental theory including the connection between the occupation and first-passage probabilities of a random walk, and the connection to electrostatics and current flows in resistor networks. The consequences of this theory are then developed for simple, illustrative geometries including the finite and semi-infinite intervals, fractal networks, spherical geometries and the wedge. Various applications are presented including neuron dynamics, self-organized criticality, diffusion-limited aggregation, the dynamics of spin systems, and the kinetics of diffusion-controlled reactions. Examples discussed include neuron dynamics, self-organized criticality, kinetics of spin systems, and stochastic resonance.

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Editorial Reviews


"...original and refreshing..." Journal of Mathematical Pyschology

"This is the first book entirely devoted to first-passage processes... Well designed and typeset, [it] is written in an easy-to-read style with a generous assortment of clearly drawn graphs. The book is very useful for anyone working in the area of stochastic processes." Mathematical Reviews

"...clearly written...the organisation and presentation of the material are excellent...a useful repository of standard and not-so-standard techniques which anyone working in the area of stochastic processes in general, and first-passage problems in particular, will want to have on their shelves." --Alan Bray, Journal of Statistical Physics

"Unquestionably a valuable book, written at an accessible level for graduate students while providing a nice summary of the last century's--and notably the last two decades'--developments of these methods. It fills a hole in the literature that's needed filling for at least ten years. Moreover, the author's style is relaxed and crystal clear while maintaining mathematical precision and power." --Charles Doering, University of Michigan

"to practitioners in the field of first- passage problems, and to students entering the field...I can recommend it strongly. It is clearly written, and the organisation and presentation of the material are excellent. It serves as a useful repository of standard and not-so-standard techniques which anyone working in the area of stochastic process in general, and first-passage problems in particular, will want to have on their shelves." Alan J. Bray, Dept of Physics and Astronomy, University of Manchester, UK

"Redner's approach is always remarkably clear and it is often aimed to develop intuition....The book is explicitly intended for allowing those with a modest background to learn essential results quickly. This goal intrinsically places it on the border between the category of textbooks and that of reference books. The author's style, colloquial and concise, yet precise, is definately appropriate for the purpose." Paolo Laureti, Econophysics

"The book is very well written and provides clear explanations of the techniques used to determine first passage probabilities and related quantities, under a variety of circumstances...this book [is] highly recommended to anyone interested in its subject, both for its clarity of presentation and for the wide range of problems treated." J.R. Dorfman, American Journal of Physics

Book Description

First-passage properties underlie a wide range of stochastic processes, such as diffusion-controlled reactions, diffusion-limited growth, neuron firing, and the triggering of stock options. This book is about the basic theory and consequences of first-passage processes. This is outlined from a modern physics perspective which emphasizes intuition, problem solving, and the application of first-passage processes in fields as diverse as random walks and resistor networks, neuron dynamics, and self-organized criticality. This variety of applications will appeal to graduate students and researchers in physics, chemistry, theoretical biology, electrical engineering, chemical engineering, operations research, and finance.

Product Details

  • Hardcover: 328 pages
  • Publisher: Cambridge University Press (August 6, 2001)
  • Language: English
  • ISBN-10: 0521652480
  • ISBN-13: 978-0521652483
  • Product Dimensions: 9.1 x 5.9 x 0.8 inches
  • Shipping Weight: 1.2 pounds (View shipping rates and policies)
  • Average Customer Review: 3.5 out of 5 stars  See all reviews (4 customer reviews)
  • Amazon Best Sellers Rank: #3,169,946 in Books (See Top 100 in Books)

More About the Author

Sid Redner is a Professor of Physics at Boston University. His research interests include statistical physics, first-passage processes, chemical kinetics, percolation theory and disordered systems, the dynamics of social systems, and the structure of complex networks. Dr. Redner has been a visiting scientist at the Schlumberger Research Center in 1984 and 1986, the Ulam Scholar at Los Alamos National Laboratory in 2004-5, an external faculty member at the Santa Fe Institute since 2007, and has held visiting professorships and University Paul Sabatier (Toulouse, France) and at University Pierre et Marie Curie (Paris, France) in 2008. Dr. Redner is a Fellow of the American Physical Society and serves on the editorial boards of several journals.

Customer Reviews

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2 of 2 people found the following review helpful
4.0 out of 5 stars an engaging survey of first-passage processes June 10, 2010
By T. Hogg
Format:Paperback|Verified Purchase
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.
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1 of 3 people found the following review helpful
2.0 out of 5 stars If this book's accurate, it's by accident January 6, 2012
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.
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3 of 7 people found the following review helpful
5.0 out of 5 stars Truly a great teacher July 30, 2002
By A Customer
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.
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3 of 8 people found the following review helpful
3.0 out of 5 stars Gamblers' ruin problems... September 15, 2005
... 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!
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