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We construct the analogue of the local time -- at a fixed point $x$ -- forMarkov processes indexed by Levy trees. We start by proving that Markovprocesses indexed by Levy trees satisfy a special Markov property which can bethought as a spatial version of the classical Markov property. Then, weconstruct the analogue of the local time by an approximation procedure and wecharacterize the support of its Lebesgue-Stieltjes measure. We also give anequivalent construction in terms of a special family of exit local times.Finally, combining these results, we show that the points at which the Markovprocess takes the value $x$ encode a new Levy tree and we construct explicitlyits height process. In particular, we recover a recent result of Le Gallconcerning the subordinate tree of the Brownian tree where the subordinationfunction is given by the past maximum process of Brownian motion indexed by theBrownian tree.

The structure of the local time of Markov processes indexed by Levy trees