Asymptotic Analysis for the Narrow Escape Problem

Abstract

Consider a domain whose boundary is reflecting, except for a finite number of absorbing sets (“windows”) with small radius. The narrow escape problem is concerned with estimating the expected time that it takes for a Brownian particle initially inside the domain to reach one of the windows and thus exit from the domain; when the particle hits the boundary at any point which is not in any of the windows, it is reflected back into the domain. The problem is motivated, for example, by a protein, or any other molecule in the cytoplasm of a cell, which moves randomly in the cytoplasm until it hits the cell membrane at one of the channels that enable it to exit the cell. In this paper we derive by rigorous mathematical analysis asymptotic estimates on the expected escape time, up to the order of the size of windows.