A CONTINUUM-TREE-VALUED MARKOV PROCESS

We present a construction of a Levy 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 psi be a critical branching mechanism. We set psi(theta)(.) = psi(. + theta) - psi(theta). Let Theta = (theta(infinity), +infinity) or Theta = [theta(infinity), +infinity) be the set of values of theta for which psi(theta) is a conservative branching mechanism. The pruning procedure allows to construct a decreasing Levy-CRT-valued Markov process (tau(theta). theta is an element of Theta), such that tau(theta) has branching mechanism psi(theta). is sub-critical if theta > 0 and super-critical if theta < 0. We then consider the explosion time A of the CRT: the smallest (negative) time theta for which the continuous state branching process (CB) associated with tau(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.