## 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.

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Singer, A., Schuss, Z. & Holcman, D. Narrow Escape, Part II: The Circular Disk.
*J Stat Phys* **122**, 465–489 (2006). https://doi.org/10.1007/s10955-005-8027-5

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DOI: https://doi.org/10.1007/s10955-005-8027-5