Regenerative tree growth: Markovian embedding of fragmenters, bifurcators, and bead splitting processes

Some, but not all processes of the form Mt=exp(..t) for a pure-jump subordinator . with Laplace exponent . arise as residual mass processes of particle 1 (tagged particle) in Bertoin.s partition-valued exchangeable fragmentation processes. We introduce the notion of a Markovian embedding of M=(Mt,t.0) in a fragmentation process, and we show that for each ., there is a unique (in distribution) binary fragmentation process in which M has a Markovian embedding. The identification of the Laplace exponent .. of its tagged particle process M. gives rise to a symmetrisation operation ...., which we investigate in a general study of pairs (M,M.) that coincide up to a random time and then evolve independently. We call M a fragmenter and (M,M.) a bifurcator. For .>0, we equip the interval R1=[0,..0M.tdt] with a purely atomic probability measure .1, which captures the jump sizes of M suitably placed on R1. We study binary tree growth processes that in the nth step sample an atom (.bead.) from .n and build (Rn+1,.n+1) by replacing the atom by a rescaled independent copy of (R1,.1) that we tie to the position of the atom. We show that any such bead splitting process ((Rn,.n),n.1) converges almost surely to an .-self-similar continuum random tree of Haas and Miermont, in the Gromov.Hausdorff.Prohorov sense. This generalises Aldous.s line-breaking construction of the Brownian continuum random tree.