Narrow Escape, Part II: The Circular Disk

Abstract

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