The Markov property of local times of Brownian motion indexed by the Brownian tree

TL;DR

This paper proves that the pair of the density and its derivative of Brownian motion indexed by the Brownian tree forms a Markov process, using excursion theory, with implications for super-Brownian motion.

Contribution

It establishes the Markov property of the density and its derivative for Brownian motion indexed by the Brownian tree, a novel result in this context.

Findings

The density process is not Markov, but the pair with its derivative is.

A similar Markov property is shown for local times of super-Brownian motion.

Methods rely on excursion theory for Brownian motion indexed by the Brownian tree.

Abstract

We consider the model of Brownian motion indexed by the Brownian tree, which has appeared in a variety of different contexts in probability, statistical physics and combinatorics. For this model, the total occupation measure is known to have a continuously differentiable density. Although the density process indexed by nonnegative reals is not Markov, we prove that the pair consisting of the density and its derivative is a time-homogeneous Markov process. We also establish a similar result for the local times of one-dimensional super-Brownian motion. Our methods rely on the excursion theory for Brownian motion indexed by the Brownian tree.