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2009 Wiener Process with Reflection in Non-Smooth Narrow Tubes
Konstantinos Spiliopoulos
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Electron. J. Probab. 14: 2011-2037 (2009). DOI: 10.1214/EJP.v14-691

Abstract

Wiener process with instantaneous reflection in narrow tubes of width $\epsilon\ll 1$ around axis $x$ is considered in this paper. The tube is assumed to be (asymptotically) non-smooth in the following sense. Let $V^{\epsilon}(x)$ be the volume of the cross-section of the tube. We assume that $\frac{1}{\epsilon}V^{\epsilon}(x)$ converges in an appropriate sense to a non-smooth function as $\epsilon\downarrow 0$. This limiting function can be composed by smooth functions, step functions and also the Dirac delta distribution. Under this assumption we prove that the $x$-component of the Wiener process converges weakly to a Markov process that behaves like a standard diffusion process away from the points of discontinuity and has to satisfy certain gluing conditions at the points of discontinuity.

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Konstantinos Spiliopoulos. "Wiener Process with Reflection in Non-Smooth Narrow Tubes." Electron. J. Probab. 14 2011 - 2037, 2009. https://doi.org/10.1214/EJP.v14-691

Information

Accepted: 28 September 2009; Published: 2009
First available in Project Euclid: 1 June 2016

zbMATH: 1195.60111
MathSciNet: MR2550291
Digital Object Identifier: 10.1214/EJP.v14-691

Subjects:
Primary: 60J60
Secondary: 37A50 , 60J99

Keywords: Delay , gluing conditions , Narrow Tubes , Non-smooth Boundary , reflection , Wiener process

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