Narrow Escape, Part II: The circular disk

Abstract

We consider Brownian motion in a circular disk $Ω$, whose boundary $\pΩ$ is reflecting, except for a small arc, $\pΩ_a$, which is absorbing. As $ε=|\partial Ω_a|/|\partial Ω|$ decreases to zero the mean time to absorption in $\pΩ_a$, denoted $Eτ$, becomes infinite. The narrow escape problem is to find an asymptotic expansion of $Eτ$ for $ε\ll1$. 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, $Eτ= \ds{\frac{|Ω|}{Dπ}}[\log\ds{\frac{1}ε}+O(1)],$ ($D$ 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 $E[τ| \x(0)=\mb{0}]=\ds{\frac{R^2}{D}}[\log\ds{\frac{1}ε} + \log 2 +\ds{1/4} + O(ε)]$. The second term in the expansion is needed in real life applications, such as trafficking of receptors on neuronal spines, because $\log\ds{\frac{1}ε}$ is not necessarily large, even when $ε$ is small. We also find the singular behavior of the probability flux profile into $\pΩ_a$ at the endpoints of $\pΩ_a$, and find the value of the flux near the center of the window.

This is the second in a series of three papers