On a Brownian motion with a hard membrane

TL;DR

This paper introduces a non-Markov process derived from Brownian motion with local perturbations, exhibiting alternating reflected behaviors on positive and negative half lines based on local time thresholds.

Contribution

It presents a novel limit process combining reflected Brownian motions with sign changes triggered by local time levels, extending classical Brownian motion models.

Findings

The process alternates between positive and negative reflections.

Behavior is governed by exponential local time thresholds.

The model captures complex boundary interactions in stochastic processes.

Abstract

Local perturbations of a Brownian motion are considered. As a limit we obtain a non-Markov process that behaves as a reflected Brownian motion on the positive half line until its local time at zero reaches some exponential level, then changes a sign and behaves as a reflected Brownian motion on the negative half line until some stopping time, etc.