Survey and Review

Abstract

The Survey and Review article in this issue is “The Narrow Escape Problem,” by D. Holcman and Z. Schuss. We are all familiar with the jagged plots formed by share prices and interest rates as they fluctuate over time. It is natural to ask questions like -when will my portfolio of Facebook shares reach $1 Million? -when will my bank's interest rate leave the 2% -- 3% range? These questions concern hitting times. Modeling the evolution as a stochastic process (a random variable that changes over time), we want to know the first occasion when the value reaches a specified level. More generally, for processes in a higher dimension, such as Robert Brown's pollen particles suspended in a fluid, we could ask when a tagged particle will first meet a specified region on the boundary of a given domain. The hitting time is itself a random variable, and we can reduce to a simpler deterministic quantity by asking for its expected value. It will then be no surprise to many readers that this deterministic value can be obtained by solving an appropriate partial differential equation. The topic of this article is a commonly arising special case which has wide practical significance and offers the added attraction of allowing mathematicians to introduce a small parameter. The narrow escape time (NET) relates to the first time that Brownian motion meets a small absorbing window on the boundary of a domain that is otherwise reflecting. Figure 1.3 illustrates the setting very clearly. A closely related problem is to determine when a particle traverses a narrow passage between almost disconnected regions, as illustrated in Figure 1.1. The article addresses the development of asymptotic methods that deal with various geometries, motivated by applications in cell biology. The underlying mathematical challenge can be viewed as solving a mixed Neumann--Dirichlet boundary value problem for the Poisson equation in a bounded domain, and the authors survey a range of developments in nonstandard asymptotics that have been brought to bear in recent years. This article will be a valuable resource for readers with interests at the interface between probability theory and partial differential equations, and those who study asymptotic analysis of singular perturbation problems and applications to physics, biochemistry, and neuroscience.