Wiener Process with Reflection in Non-Smooth Narrow Tubes

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 V (x) be the volume of the cross-section of the tube. We assume that 1 ε V (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.