Squared Bessel processes of positive and negative dimension embedded in Brownian local times

TL;DR

This paper embeds squared Bessel processes of all real dimensions into Brownian local times by decomposing Brownian paths via excursions, linking to recent frameworks involving point processes and continuum tree evolutions.

Contribution

It introduces a novel embedding of all real-dimension squared Bessel processes into Brownian local times using excursion decomposition and the Harrison--Shepp skew Brownian motion construction.

Findings

Embedded all real-dimension squared Bessel processes in Brownian local times.

Connected Bessel process embeddings to point processes of excursions and stable processes.

Linked the framework to continuum tree evolutions and the Aldous diffusion.

Abstract

The Ray--Knight theorems show that the local time processes of various path fragments derived from a one-dimensional Brownian motion $B$ are squared Bessel processes of dimensions $0$, $2$, and $4$. It is also known that for various singular perturbations $X= |B| + \mu \ell$ of a reflecting Brownian motion $|B|$ by a multiple $\mu$ of its local time process $\ell$ at $0$, corresponding local time processes of $X$ are squared Bessel with other real dimension parameters, both positive and negative. Here, we embed squared Bessel processes of all real dimensions directly in the local time process of $B$. This is done by decomposing the path of $B$ into its excursions above and below a family of continuous random levels determined by the Harrison--Shepp construction of skew Brownian motion as the strong solution of an SDE driven by $B$. This embedding connects to Brownian local times a framework of point processes of squared Bessel excursions of negative dimension and associated stable processes, recently introduced by Forman, Pal, Rizzolo and Winkel to set up interval partition evolutions that arise in their approach to the Aldous diffusion on a space of continuum trees.