Narrow Escape, Part II: The Circular Disk

We consider Brownian motion in a circular disk Ω, whose boundary is reflecting, except for a small arc, , which is absorbing. As decreases to zero the mean time to absorption in , denoted , becomes infinite. The narrow escape problem is to find an asymptotic expansion of for . 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, ( 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 . The second term in the expansion is needed in real life applications, such as trafficking of receptors on neuronal spines, because is not necessarily large, even when ε is small. We also find the singular behavior of the probability flux profile into at the endpoints of , and find the value of the flux near the center of the window.