D ec 2 00 4 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, ∂Ωa, which is absorbing. As ε = |∂Ωa|/|∂Ω| decreases to zero the mean time to absorption in ∂Ωa, 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 smooth domain, Eτ = |Ω| Dπ [