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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&#39;s theorem. We also extend the pruning procedure to this super-critical case. Let $ψ$ be a critical branching mechanism. We set $ψ_θ(\cdot)=ψ(\cdot+θ)-ψ(θ)$. Let $Θ=(θ_\infty,+\infty)$ or $Θ=[θ_\infty,+\infty)$ be the set of values of $θ$ for which $ψ_θ$ is a branching mechanism. The pruning procedure allows to construct a decreasing Lévy-CRT-valued Markov process $(\ct_θ,θ\inΘ)$, such that $\mathcal{T}_θ$ 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 smaller (negative) time $θ$ for which $\mathcal{T}_θ$ has finite mass. 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 CRT 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&#39; CRT.