Asymptotic analysis of the Narrow Escape Problem in general shaped domain with several absorbing necks

Abstract: This paper considers the two-dimensional narrow escape problem in a domain which is composed of a relatively big head and several thin necks. The narrow escape problem is to compute the mean first passage time(MFPT) of a Brownian particle traveling from inside the head to the end of the necks. The original model for MFPT is to solve a mixed Dirichlet-Neumann boundary value problem for the Poisson equation in the composite domain, and is computationally challenging. In this paper, we compute the MFPT by solving an equivalent Neumann-Robin type boundary value problem. By solving the new model, we obtain the high order asymptotic expansion of the MFPT. We also conduct numerical experiments to show the accuracy of the high order expansion. As far as we know, this is the first result on high order asymptotic solution for NEP in a general shaped domain with several absorbing neck windows. This work is motivated by \cite{Li}, where the Neumann-Robin model was proposed to solve the NEP in a domain with a single absorbing neck.