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Narrow Escape in \({\mathbb{R}}^{3}\)

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Book cover Brownian Dynamics at Boundaries and Interfaces

Part of the book series: Applied Mathematical Sciences ((AMS,volume 186))

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Abstract

The NET problem in three dimensions is more complicated than that in two dimension, primarily because the singularity of Neumann’s function for a regular domain is more complicated than (7.1).

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Notes

  1. 1.

    For nonsmooth p 0, the integral is not uniformly bounded. For example, for \(p_{0} =\delta (\mbox{ $\boldsymbol{x}$} -\mbox{ $\boldsymbol{x}$}_{0})\), we have \(\int _{\Omega }N(\mbox{ $\boldsymbol{x}$},\mbox{ $\boldsymbol{y}$})p_{0}(\mbox{ $\boldsymbol{y}$})\,\mathrm{d}\mbox{ $\boldsymbol{y}$} = N(\mbox{ $\boldsymbol{x}$},\mbox{ $\boldsymbol{x}$}_{0})\), which becomes singular as \(\mbox{ $\boldsymbol{x}$} \rightarrow \mbox{ $\boldsymbol{x}$}_{0}\). However, this is an integrable singularity, and as such it does not affect the leading-order asymptotics in δ.

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  1. School of Mathematical Sciences, Tel Aviv University, Tel Aviv, Israel

    Zeev Schuss

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Schuss, Z. (2013). Narrow Escape in \({\mathbb{R}}^{3}\) . In: Brownian Dynamics at Boundaries and Interfaces. Applied Mathematical Sciences, vol 186. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-7687-0_8

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