Book contents
- Frontmatter
- Contents
- Preface
- Errata
- 1 First-Passage Fundamentals
- 2 First Passage in an Interval
- 3 Semi-Infinite System
- 4 Illustrations of First Passage in Simple Geometries
- 5 Fractal and Nonfractal Networks
- 6 Systems with Spherical Symmetry
- 7 Wedge Domains
- 8 Applications to Simple Reactions
- References
- Index
1 - First-Passage Fundamentals
Published online by Cambridge University Press: 05 August 2012
- Frontmatter
- Contents
- Preface
- Errata
- 1 First-Passage Fundamentals
- 2 First Passage in an Interval
- 3 Semi-Infinite System
- 4 Illustrations of First Passage in Simple Geometries
- 5 Fractal and Nonfractal Networks
- 6 Systems with Spherical Symmetry
- 7 Wedge Domains
- 8 Applications to Simple Reactions
- References
- Index
Summary
What Is a First-Passage Process?
This book is concerned with the first-passage properties of random walks and diffusion, and the basic consequences of first-passage phenomena. Our starting point is the first-passage probability; this is the probability that a diffusing particle or a random-walk first reaches a specified site (or set of sites) at a specified time. The importance of first-passage phenomena stems from its fundamental role in stochastic processes that are triggered by a first passage event. Typical examples include fluorescence quenching, in which light emission by a fluorescent molecule stops when it reacts with a quencher; integrate-and-fire neurons, in which a neuron fires only when a fluctuating voltage level first reaches a specified level; and the execution of buy/sell orders when a stock price first reaches a threshold. Further illustrations are provided throughout this book.
A Simple Illustration
To appreciate the essential features of first-passage phenomena, we begin with a simple example. Suppose that you are a nervous investor who buys stock in a company at a price of $100. Suppose also that this price fluctuates daily in a random multiplicative fashion. That is, at the end of each day the stock price changes by a multiplicative factor f < 1 or by f−1 compared with the previous day's price, with each possibility occurring with probability 1/2 (Fig. 1.1). The multiplicative change ensures that the price remains positive. To be concrete, let's take f = 90% and suppose that there is a loss on the first day so that the stock price drops to $90.
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- A Guide to First-Passage Processes , pp. 1 - 37Publisher: Cambridge University PressPrint publication year: 2001
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