A continuum-tree-valued Markov process

We present a construction of a Lévy continuum random tree (CRT) associated with a super-critical continuous state branching process using the so-called exploration process and a Girsanov theorem. We also extend the pruning procedure to this super-critical case. Let Ψ be a critical branching mechanism. We set Ψ _θ(̇) = Ψ(̇ + θ) - Ψ(θ). Let Θ = (θ _∞,+∞) or Θ = [θ _∞,+∞) be the set of values of θ for which Ψ _θ is a conservative branching mechanism. The pruning procedure allows to construct a decreasing Lévy-CRT-valued Markov process (Tθ _θ Ψθ), such that T _θ has branching mechanism Ψ _θ. It is sub-critical if θ >0 and super-critical if θ <0. We then consider the explosion time A of the CRT: the smallest (negative) time A for which the continuous state branching process (CB) associated with T _θ has finite total mass (i.e., the length of the excursion of the exploration process that codes the CRT is finite). We describe the law of A as well as the distribution of the CRT just after this explosion time. The CRT just after explosion can be seen as a CRT conditioned not to be extinct which is pruned with an independent intensity related to A. We also study the evolution of the CRT-valued process after the explosion time. This extends results from Aldous and Pitman on Galton-Watson trees. For the particular case of the quadratic branching mechanism, we show that after explosion the total mass of the CB behaves like the inverse of a stable subordinator with index 1/2. This result is related to the size of the tagged fragment for the fragmentation of Aldous's CRT.