The Markov property of local times of 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 ${({\ell }^{\mathit{x}})}_{\mathit{x}\in \mathbb{R}}$, and we write ${({\dot{\ell }}^{\mathit{x}})}_{\mathit{x}\in \mathbb{R}}$ for its derivative. Although the process ${({\ell }^{\mathit{x}})}_{\mathit{x}\ge 0}$ is not Markov, we prove that the pair ${({\ell }^{\mathit{x}},{\dot{\ell }}^{\mathit{x}})}_{\mathit{x}\ge 0}$ 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.