Resumen de Regenerative tree growth: Markovian embedding of fragmenters, bifurcators, and bead splitting processes
Some, but not all processes of the form Mt=exp(−ξt) for a pure-jump subordinator ξ with Laplace exponent Φ 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 Φ, there is a unique (in distribution) binary fragmentation process in which M has a Markovian embedding. The identification of the Laplace exponent Φ∗ of its tagged particle process M∗ gives rise to a symmetrisation operation Φ↦Φ∗, 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 α>0, we equip the interval R1=[0,∫∞0Mαtdt] with a purely atomic probability measure μ1, 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 μn and build (Rn+1,μn+1) by replacing the atom by a rescaled independent copy of (R1,μ1) that we tie to the position of the atom. We show that any such bead splitting process ((Rn,μn),n≥1) converges almost surely to an α-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.
