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. 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.

arXiv commentRevised version, taking account of all remarks made by referees. In particular, an error at the end of the proof of Lemma 8 has been corrected. A couple of arguments have also been simplified