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A Guide to First-Passage Processes, by Sidney Redner
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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.
- Sales Rank: #316842 in Books
- Published on: 2007-06-21
- Original language: English
- Number of items: 1
- Dimensions: 8.98" h x .75" w x 5.98" l, 1.06 pounds
- Binding: Paperback
- 328 pages
Review
"...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
About the Author
Sid Redner is a condensed-matter theorist whose research focuses on non-equilibrium statistical physics and its applications. Dr Redner has been on the physics faculty at Boston University since 1978 and has been a full professor since 1989. He has published 230 research articles and is the author of A Kinetic View of Statistical Physics with P. L. Krapivsky and E. Ben-Naim (Cambridge University Press, 2010). Dr Redner is a Fellow of the American Physical Society and was a visiting scientist at Schlumberger Research in 1984-1985, the Ulam Scholar at Los Alamos National Laboratory in 2004-2005 and a visiting professor at the Universite Paul Sabatier (Toulouse) and Universite Pierre et Marie Curie (Paris) in 2008.
Most helpful customer reviews
3 of 8 people found the following review helpful.
Gamblers' ruin problems...
By Professor Joseph L. McCauley
... 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!
2 of 3 people found the following review helpful.
an engaging survey of first-passage processes
By T. Hogg
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.
4 of 8 people found the following review helpful.
Truly a great teacher
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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