Abstract
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 on the tree, using L'evy snake techniques. We then prove that the resulting sub-tree after pruning is still a L'evy continuum random tree. This last result is proved using the exploration process that codes the CRT, 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.
Highlights
Continuous state branching processes (CSBP) were first introduced by Jirina [25] and it is known since Lamperti [27] that these processes are the scaling limits of Galton-Watson processes
Let us recall that α represents a drift term, β is a diffusion coefficient and π describes the jumps of the CSBP
The goal of this paper is to introduce a general pruning procedure of a genealogical tree associated with a branching mechanism ψ of the form (1), which is the continuous analogue of the previous percolation
Summary
Continuous state branching processes (CSBP) were first introduced by Jirina [25] and it is known since Lamperti [27] that these processes are the scaling limits of Galton-Watson processes. We consider the Levy CRT associated with a general critical or sub-critical branching mechanism ψ (or rather the exploration process that codes that tree) and we add marks on the tree. As we don’t use the real trees framework but only the exploration processes that codes the Levy CRTs, we must describe all these marks in term of exploration processes. We define the measure mstke as the derivative of the function Wt. Let us remark that in [22], the height process is supposed to be continuous for the construction of Levy snakes. We denote by At the Lebesgue measure of the set of the individuals prior to t whose lineage does not contain any mark i.e. We consider its right-continuous inverse Ct := inf{r ≥ 0, Ar > t} and we define the pruned exploration process ρby. The Appendix is devoted to some extension of the Levy snake when the height process is not continuous
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