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.
Highlights
For each x ∈ and 0 < ε
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 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
For the convenience of the reader, we briefly recall the Feller characterization of all one-dimensional Markov processes, that are continuous with probability one
Summary
For each x ∈ and 0 < ε
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