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

Abstract

Some, but not all processes of the form $M_{t}=\exp(-\xi_{t})$ for a pure-jump subordinator $\xi$ with Laplace exponent $\Phi$ 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 $M=(M_{t},t\ge0)$ in a fragmentation process, and we show that for each $\Phi$, there is a unique (in distribution) binary fragmentation process in which $M$ has a Markovian embedding. The identification of the Laplace exponent $\Phi^{*}$ of its tagged particle process $M^{*}$ gives rise to a symmetrisation operation $\Phi\mapsto\Phi^{*}$, 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 $\alpha>0$, we equip the interval $R_{1}=[0,\int_{0}^{\infty}M_{t}^{\alpha}\,dt]$ with a purely atomic probability measure $\mu_{1}$, 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 $\mu_{n}$ and build $(R_{n+1},\mu_{n+1})$ by replacing the atom by a rescaled independent copy of $(R_{1},\mu_{1})$ that we tie to the position of the atom. We show that any such bead splitting process $((R_{n},\mu_{n}),n\ge1)$ converges almost surely to an $\alpha$-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.