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Wiener process with reflection in non-smooth narrow tubes. (English) Zbl 1195.60111

Summary: A Wiener process with instantaneous reflection in narrow tubes of width \(\varepsilon \ll 1\) around axis \(x\) is considered. The tube is assumed to be (asymptotically) non-smooth in the following sense. Let \(V^{\varepsilon}(x)\) be the volume of the cross-section of the tube. We assume that \((1/\varepsilon)V^{\varepsilon}(x)\) converges in an appropriate sense to a non-smooth function as \(\varepsilon \downarrow 0\). 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.

MSC:

60J65 Brownian motion
60J99 Markov processes
37A50 Dynamical systems and their relations with probability theory and stochastic processes
60F17 Functional limit theorems; invariance principles
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