caa a22 4500 467892539 CHVBK 20180406152753.0 cr unu---uuuuu 170328e20060201xx s 000 0 eng 10.1007/s10955-005-8027-5 doi (NATIONALLICENCE)springer-10.1007/s10955-005-8027-5 Narrow Escape, Part II: The Circular Disk [Elektronische Daten] [A. Singer, Z. Schuss, D. Holcman] 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. Springer Science + Business Media, Inc., 2006 Planar Brownian motion nationallicence Exit problem nationallicence Singular perturbations nationallicence Singer A. Department of Mathematics, Yale University, 10 Hillhouse Ave., PO Box 208283, 06520-8283, New Haven, CT, USA aut Schuss Z. Department of Mathematics, Tel-Aviv University, 69978, Tel-Aviv, Israel aut Holcman D. Department of Mathematics, Weizmann Institute of Science, 76100, Rehovot, Israel aut Journal of Statistical Physics Kluwer Academic Publishers-Plenum Publishers 122/3(2006-02-01), 465-489 0022-4715 122:3<465 2006 122 10955 https://doi.org/10.1007/s10955-005-8027-5 text/html Onlinezugriff via DOI 1 research-article jats NATIONALLICENCE 856 40 https://doi.org/10.1007/s10955-005-8027-5 text/html Onlinezugriff via DOI NATIONALLICENCE 700 1- Singer A. Department of Mathematics, Yale University, 10 Hillhouse Ave., PO Box 208283, 06520-8283, New Haven, CT, USA aut NATIONALLICENCE 700 1- Schuss Z. Department of Mathematics, Tel-Aviv University, 69978, Tel-Aviv, Israel aut NATIONALLICENCE 700 1- Holcman D. Department of Mathematics, Weizmann Institute of Science, 76100, Rehovot, Israel aut NATIONALLICENCE 773 0- Journal of Statistical Physics Kluwer Academic Publishers-Plenum Publishers 122/3(2006-02-01), 465-489 0022-4715 122:3<465 2006 122 10955 Metadata rights reserved Springer special CC-BY-NC licence nationallicence BK010053 XK010053 XK010000 NATIONALLICENCE NATIONALLICENCE NL-springer