The Narrow Escape Problem

Narrow-escape-time problem: the imperfect trapping case.

We present a master equation approach to the narrow escape time (NET) problem, i.e., the time needed for a particle contained in a confining domain with a single narrow opening to exit the domain for the first time. We introduce a finite transition probability, ν, at the narrow escape window, allowing the study of the imperfect trapping case. Ranging from 0 to ∞, ν allowed the study of both ext...

Oscillatory Survival Probability: Analytical, Numerical Study for oscillatory narrow escape and applications to neural network dynamics

We study the escape of Brownian motion from the domain of attraction Ω of a stable focus with a strong drift. The boundary ∂Ω of Ω is an unstable limit cycle of the drift and the focus is very close to the limit cycle. We find a new phenomenon of oscillatory decay of the peaks of the survival probability of the Brownian motion in Ω. We compute explicitly the complex-valued second eigenvalue λ2(...

On the narrow escape problem

Narrow escape and leakage of Brownian particles.

Questions of flux regulation in biological cells raise a renewed interest in the narrow escape problem. The determination of a higher order asymptotic expansion of the narrow escape time depends on determining the singularity behavior of the Neumann Green's function for the Laplacian in a three-dimensional (3D) domain with a Dirac mass on the boundary. In addition to the usual 3D Coulomb singul...

The Mean Escape Time for a Narrow Escape Problem with Multiple Switching Gates

This paper deals with the narrow escape problem when there are two gates which open alternatively in a random way. We set up the problem and perform rigorous asymptotic analysis to derive the mean escape time (MET) for a Brownian particle inside a domain to exit the domain through switching gates. We show that the leading order term of the asymptotic expansion of the MET is twice the leading or...

Particle lifetime in cylindrical cavity with absorbing spot on the wall: Going beyond the narrow escape problem.

The mean lifetime of a particle diffusing in a cylindrical cavity with a circular absorbing spot on the cavity wall is studied analytically as a function of the spot radius, its location on the wall, the particle initial position, and the cavity shape determined by its length and radius. When the spot radius tends to zero our formulas for the mean lifetime reduce to the result given by the solu...