The narrow escape problem – a short review of recent results. (English) Zbl 1259.82083
The mean first passage time or narrow escape time of a particle from a point \(x\) in a bounded domain to the absorbing boundary is understood as being the mean of the solution of a mixed Dirichlet-Neumann boundary value problem for the Poisson equation. The Narrow Escape Problem (NET) consists of calculating this mean. In the literature, contributions on three main classes of this NET problem are known: absorbing patches on a smooth boundary (class1), absorbing patches at the end of a smooth funnel (class2), patches at the end of a long narrow neck attached smoothly or non-smoothly to the reflecting boundary (class3). In the present article, one reviews – shortly – some recent results for domains of these classes, in two as well in as in three dimensions. To this very short paper, references of 41 titles are attached.
Reviewer: Claudia Simionescu-Badea (Wien)
MSC:
| 82C32 | Neural nets applied to problems in time-dependent statistical mechanics |
| 35J05 | Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation |
| 35J25 | Boundary value problems for second-order elliptic equations |
| 60J65 | Brownian motion |
| 58J65 | Diffusion processes and stochastic analysis on manifolds |
Keywords:
Brownian motion; mean first passage time; singular perturbations; Poisson equation; Dirichlet-Neumann boundary value problemReferences:
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