@article{EJP691,
author = {Konstantinos Spiliopoulos},
title = {Wiener Process with Reflection in Non-Smooth Narrow Tubes},
journal = {Electron. J. Probab.},
fjournal = {Electronic Journal of Probability},
volume = {14},
year = {2009},
keywords = {Narrow Tubes; Wiener Process; Reflection; Non-smooth Boundary; Gluing Conditions; Delay},
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.},
pages = {no. 69, 2011-2037},
issn = {1083-6489},
doi = {10.1214/EJP.v14-691},
url = {http://ejp.ejpecp.org/article/view/691}}