Wiener Process with Reflection in Non-Smooth Narrow Tubes

TL;DR

This paper studies the limiting behavior of a Wiener process reflected in narrow, non-smooth tubes, showing convergence to a Markov process with specific boundary conditions at discontinuities.

Contribution

It introduces a framework for analyzing Wiener processes in non-smooth narrow tubes and establishes their weak convergence to a Markov process with gluing conditions.

Findings

The x-component converges to a diffusion process away from discontinuities.

The limiting process satisfies specific gluing conditions at points of non-smoothness.

The approach handles tubes with volume functions converging to non-smooth limits.

Abstract

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.