Meeting of squared Bessel flow lines and application to the skew Brownian motion

TL;DR

This paper investigates the interaction of squared Bessel flow lines, introduces a jump Markov process from their meeting levels, and applies these findings to extend results on skew Brownian motion flows and their exceptional times.

Contribution

It establishes a connection between squared Bessel flow line meetings and jump Markov processes, extending existing skew Brownian motion flow results and analyzing bifurcation times.

Findings

Meeting levels form a jump Markov process.

Extended local time flow results for skew Brownian motions.

Computed Hausdorff dimension of bifurcation times.

Abstract

We study the meeting level between squared Bessel (BESQ) flow lines of different dimensions, and show that it gives rise to a jump Markov process. We apply these results to the skew Brownian flow introduced by Burdzy and Chen \cite{burdzy2001local} and Burdzy and Kaspi \cite{burdzy2004lenses}. It allows us to extend the results of \cite{burdzy2001local} and of Gloter and Martinez \cite{gloter2013distance} describing the local time flow of skew Brownian motions. Finally, we compute the Hausdorff dimension of exceptional times revealed by Burdzy and Kaspi \cite{burdzy2004lenses} when skew Brownian flow lines bifurcate.