  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.
