Narrow Escape, Part II: The circular disk

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dc.creatorSinger, A.
dc.creatorSchuss, Z.
dc.creatorHolcman, D.
dc.date2004-12-15
dc.date.accessioned2026-07-07T04:31:44Z
dc.date.available2026-07-07T04:31:44Z
dc.descriptionWe 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.
dc.descriptionThis is the second in a series of three papers
dc.identifierhttps://arxiv.org/abs/math-ph/0412050
dc.identifierhttp://arxiv.org/abs/math-ph/0412050
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/57929
dc.subjectMathematical Physics
dc.subjectProbability
dc.subject60J65; 58J05; 58J32; 58J37
dc.titleNarrow Escape, Part II: The circular disk
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