Narrow escape, part II: The circular disk

We consider Brownian motion in a circular disk Ω, whose boundary ∂Ω is reflecting, except for a small arc, ∂Ω_a, which is absorbing. As ε = |∂Ω_a|/|∂Ω decreases to zero the mean time to absorption in ∂Ω_a, denoted Eτ, becomes infinite. The narrow escape problem is to find an asymptotic expansion of Eτ for ε ≪ 1. We find the first two terms in the expansion and an estimate of the error. The results are extended in a straightforward manner to planar domains and two-dimensional Riemannian manifolds that can be mapped conformally onto the disk. Our results improve the previously derived expansion for a general domain, Eτ = |Ω|/Dπ[log 1/ε + O(1)], (D is the diffusion coefficient) in the case of a circular disk. We find that the mean first passage time from the center of the disk is E[τ|x(0) = 0] = R²/D[log 1/ε + log 2 + 1/4 + Oε]. The second term in the expansion is needed in real life applications, such as trafficking of receptors on neuronal spines, because log 1/ε is not necessarily large, even when ε is small. We also find the singular behavior of the probability flux profile into ∂Ω_a at the endpoints of ∂Ω_a, and find the value of the flux near the center of the window.