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 Electronic Journal of Probability > Vol. 14 (2009) > Paper 69 open journal systems 


Wiener Process with Reflection in Non-Smooth Narrow Tubes

Konstantinos Spiliopoulos, Brown University


Abstract
Wiener process with instantaneous reflection in narrow tubes of width ε<<1 around axis x is considered in this paper. The tube is assumed to be (asymptotically) non-smooth in the following sense. Let Vε(x) be the volume of the cross-section of the tube. We assume that (1/ε)Vε(x) converges in an appropriate sense to a non-smooth function as ε->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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Pages: 2011-2037

Published on: September 28, 2009
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Electronic Journal of Probability. ISSN: 1083-6489