Narrow Escape, Part I

A Brownian particle with diffusion coefficient D is confined to a bounded domain Ω by a reflecting boundary, except for a small absorbing window $$\partial\Omega_a$$. The mean time to absorption diverges as the window shrinks, thus rendering the calculation of the mean escape time a singular perturbation problem. In the three-dimensional case, we construct an asymptotic approximation when the window is an ellipse, assuming the large semi axis a is much smaller than $$|\Omega|^{1/3}$$ ($$|\Omega|$$ is the volume), and show that the mean escape time is $$E\tau\sim{\frac{|\Omega|}{2\pi Da}} K(e)$$, where e is the eccentricity and $$K(\cdot)$$ is the complete elliptic integral of the first kind. In the special case of a circular hole the result reduces to Lord Rayleigh's formula $$E\tau\sim{\frac{|\Omega|}{4aD}}$$, which was derived by heuristic considerations. For the special case of a spherical domain, we obtain the asymptotic expansion $$E\tau={\frac{|\Omega|}{4aD}} [1+\frac{a}{R} \log \frac{R}{a} + O(\frac{a}{R})]$$. This result is important in understanding the flow of ions in and out of narrow valves that control a wide range of biological and technological function. If Ω is a two-dimensional bounded Riemannian manifold with metric g and $$\varepsilon=|\partial\Omega_a|_g/|\Omega|_g\ll1$$, we show that $$E\tau ={\frac{|\Omega|_g}{D\pi}}[\log{\frac{1}{\varepsilon}}+O(1)]$$. This result is applicable to diffusion in membrane surfaces.