  We construct the analogue of the local time -- at a fixed point $x$ -- for
Markov processes indexed by Levy trees. We start by proving that Markov
processes indexed by Levy trees satisfy a special Markov property which can be
thought as a spatial version of the classical Markov property. Then, we
construct the analogue of the local time by an approximation procedure and we
characterize the support of its Lebesgue-Stieltjes measure. We also give an
equivalent construction in terms of a special family of exit local times.
Finally, combining these results, we show that the points at which the Markov
process takes the value $x$ encode a new Levy tree and we construct explicitly
its height process. In particular, we recover a recent result of Le Gall
concerning the subordinate tree of the Brownian tree where the subordination
function is given by the past maximum process of Brownian motion indexed by the
Brownian tree.
