Abstract
The method of Chap. 1 obviously fails for Brownian motion in two-dimensional domains whose boundaries are smooth and reflecting, except for a small absorbing window at the end of a cusp-shaped funnel, as shown in Figs. 1.2(left) and 2.1. The cusp can be formed by a partial block of a planar domain, as shown in the schematic Fig. 2.2(left).
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Notes
- 1.
It is known in partial differential equations theory in higher dimensions that at boundary points where \(\sum _{i,j}\sigma ^{ij}(\boldsymbol{x})\nu ^{i}(\boldsymbol{x})\nu ^{j}(\boldsymbol{x}) = 0\) , boundary conditions can be imposed only at points where \(\boldsymbol{a}(\boldsymbol{x}) \cdot \boldsymbol{\nu }(\boldsymbol{x}) <0\).
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Holcman, D., Schuss, Z. (2015). Special Asymptotics for Stochastic Narrow Escape. In: Stochastic Narrow Escape in Molecular and Cellular Biology. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-3103-3_2
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