A continuum-tree-valued Markov process

Abstract

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 $({\mathcal{T}}_{\theta},\theta\in\Theta)$, such that ${\mathcal{T}}_{\theta}$ 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 θ for which the continuous state branching process (CB) associated with ${\mathcal{T}}_{\theta}$ 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.