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 ∂Ω_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 |Ω|^1/3 (|Ω| is the volume), and show that the mean escape time is Eτ ∼ |Ω|/2π Da K(e), where e is the eccentricity and K(·) 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 Eτ ∼ |Ω|/4aD, which was derived by heuristic considerations. For the special case of a spherical domain, we obtain the asymptotic expansion Eτ = |Ω|/4aD[1 + a/R log R/a + O(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 ε = |∂Ω_a|_g ≪ 1, we show that Eτ = |Ω|_g/Dπ[log 1/ε + O(1)]. This result is applicable to diffusion in membrane surfaces.