Diffusion processes on branching Brownian motion

TL;DR

This paper constructs a new class of diffusion processes on branching Brownian motion particles, linked to extremal processes and supported on a Cantor-like set, resembling Liouville Brownian motion.

Contribution

It introduces a novel diffusion process on branching Brownian motion particles, extending the concept of Liouville Brownian motion to this setting.

Findings

Processes are symmetric with respect to limits of random martingale measures.

Support of measures is on a Cantor-like set.

Analogous to Liouville Brownian motion in Gaussian free fields.

Abstract

We construct a class of one-dimensional diffusion processes on the particles of branching Brownian motion that are symmetric with respect to the limits of random martingale measures. These measures are associated with the extended extremal process of branching Brownian motion and are supported on a Cantor-like set. The processes are obtained via a time-change of a standard one-dimensional reflected Brownian motion on $\mathbb{R}_+$ in terms of the associated positive continuous additive functionals. The processes introduced in this paper may be regarded as an analogue of the Liouville Brownian motion which has been recently constructed in the context of a Gaussian free field.