Narrow Escape, Part II: The circular disk

Abstract

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