Lorentz Process with shrinking holes in a wall

TL;DR

This paper studies a Lorentz process in a strip with a shrinking hole in the wall, showing that it converges to a quasi-reflected Brownian motion, and provides local time results for the process.

Contribution

It introduces a new limit process for a Lorentz system with a shrinking hole, and derives local time properties of the process.

Findings

The scaled Lorentz process converges to a quasi-reflected Brownian motion.

Local time results for the Lorentz process are established.

The limiting process is Markovian but not strong Markovian.

Abstract

We ascertain the diffusively scaled limit of a periodic Lorentz process in a strip with an almost reflecting wall at the origin. Here, almost reflecting means that the wall contains a small hole waning in time. The limiting process is a quasi-reflected Brownian motion, which is Markovian but not strong Markovian. Local time results for the periodic Lorentz process, having independent interest, are also found and used.