Some, but not all processes of the form M t = exp(−ξ t ) for a pure-jump subordinator ξ with Laplace exponent Φ arise as residual mass processes of particle 1 (tagged particle) in an exchangeable fragmentation processes. We introduce the notion of a Markovian embedding of M in an exchangeable fragmentation process and show that for each Φ, there is a unique binary dislocation measure ν such that M has a Markovian embedding in an associated exchangeable fragmentation process. 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 junction time and then evolve independently. We call M a fragmenter and (M, M * ) a bifurcator. For all Φ and α > 0, we can represent a fragmenter M as an interval R 1 = [0, equipped with a purely atomic probability measure µ 1 capturing the jump sizes of M t after an α-self-similar time-change. We call (R 1 , µ 1 ) an (α, Φ)-string of beads. We study binary tree growth processes that in the nth step sample a bead from µ n and build (R n+1 , µ n+1 ) by splitting the bead into a new string of beads, a rescaled independent copy of (R 1 , µ 1 ) that we tie to the position of the sampled bead. We show that all such bead splitting processes converge almost surely to an α-self-similar CRT, in the Gromov-Hausdorff-Prohorov sense.
In REF we investigate the fact that not all fragmenters appear as residual mass processes of typical (uniformly random) leaves.
Regenerative tree growth: Markovian embedding of fragmenters, bifurcators, and bead splitting processes
Annals of Probability 2015, Vol. 43, No. 5, 2611-2646
Mathematics