Wiener Process with Reflection in Non-Smooth Narrow Tubes

Details Wiener Process with Reflection in Non-Smooth Narrow Tubes Wiener Process with Reflection in Non-Smooth Narrow Tubes

2010-04-18

arxiv.org/abs/1004.2991v1

Electronic Journal of Probability, Vol. 14, Paper no. 69, (2009), pp. 2011-2037

Mathematics

Probability

28 pages, 1 figure

Scientific paper

Wiener process with instantaneous reflection in narrow tubes of width {\epsilon}<<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 $V^{\epsilon}(x)/{\epsilon}$ converges in an appropriate sense to a non-smooth function as {\epsilon}->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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