The narrow escape problem

D. Holcman, Z. Schuss

Research output: Contribution to journalArticlepeer-review

Abstract

The narrow escape problem in diffusion theory is to calculate the mean first passage time of a diffusion process to a small target on the reflecting boundary of a bounded domain. The problem is equivalent to solving the mixed Dirichlet-Neumann boundary value problem for the Poisson equation with small Dirichlet and large Neumann parts. The mixed boundary value problem, which goes back to Lord Rayleigh, originates in the theory of sound and is closely connected to the eigenvalue problem for the mixed problem and for the Neumann problem in domains with bottlenecks. We review here recent developments in the nonstandard asymptotics of the problem, which are based on several ingredients: a better resolution of the singularity of Neumann's function, resolution of the boundary layer near the small target by conformal mappings of domains with bottlenecks, and the breakup of composite domains into simpler components. The new methodology applies to two- and higher-dimensional problems. Selected applications are reviewed.

Original languageEnglish
Pages (from-to)213-257
Number of pages45
JournalSIAM Review
Volume56
Issue number2
DOIs
StatePublished - 2014

Keywords

  • Asymptotic expansions
  • Diffusion processes
  • Laplace equation
  • Mean first passage time
  • Mixed Dirichlet-Neumann boundary value problem
  • Narrow escape
  • Neumann's function
  • Stochastic modeling

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