Matched asymptotic analysis to solve the narrow escape problem in a domain with a long neck

Abstract

In this study, we mainly consider the narrow escape problem (NEP) in a two-dimensional domain Ω with a long neck, which is the two-dimensional analogue of a dendritic spine geometry. The NEP requires the computation of the mean escape time of a Brownian particle starting from the head until it exits from the end of the neck, where the particle is absorbed. We divide the domain into the neck part Ωn and the head part Ωh, with the common boundary . The escape time in Ωh can be considered to be the time from the head to the end of the neck, while the escape time in Ωn can be considered to be the time from the neck to the end of the neck. We compute the two exit times separately and match them by considering some boundary value problem with an impedance boundary condition on , which we refer to as the Neumann–Robin boundary model.