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Pruning a Lévy continuum random tree. (English) Zbl 1231.60073

Summary: Given a general critical or sub-critical branching mechanism, we define a pruning procedure of the associated Lévy continuum random tree. This pruning procedure is defined by adding some marks to the tree, using Lévy snake techniques. We then prove that the resulting sub-tree after pruning is still a Lévy continuum random tree. This last result is proved using the exploration process that codes the continuum random tree, a special Markov property and martingale problems for exploration processes. We finally give the joint law under the excursion measure of the lengths of the excursions of the initial exploration process and the pruned one.

MSC:

60J25 Continuous-time Markov processes on general state spaces
60G57 Random measures
60J80 Branching processes (Galton-Watson, birth-and-death, etc.)
60J68 Superprocesses
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