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

Abstract

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 Bertoin's partition-valued exchangeable fragmentation processes. We introduce the notion of a Markovian embedding of $M=(M_t,t\ge 0)$ 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 $Φ\mapstoΦ^*$, 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 $R_1=[0,\int_0^{\infty}M_t^α\,dt]$ with a purely atomic probability measure $μ_1$, which captures the jump sizes of $M$ suitably placed on $R_1$. We study binary tree growth processes that in the $n$th step sample an atom (``bead'') from $μ_n$ and build $(R_{n+1},μ_{n+1})$ by replacing the atom by a rescaled independent copy of $(R_1,μ_1)$ that we tie to the position of the atom. We show that any such bead splitting process $((R_n,μ_n),n\ge1)$ 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.