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

TL;DR

This paper introduces a Markovian embedding framework for fragmentation processes, characterizes a symmetrization operation on Laplace exponents, and demonstrates convergence of bead splitting tree processes to self-similar continuum random trees.

Contribution

It develops a novel Markovian embedding approach for fragmentation processes, introduces a symmetrization operation on Laplace exponents, and proves convergence of bead splitting processes to self-similar trees.

Findings

Unique binary fragmentation process with Markovian embedding for each Laplace exponent.

Symmetrization operation on Laplace exponents and its properties.

Almost sure convergence of bead splitting processes to self-similar continuum random trees.

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 $M=(M_t,t\ge 0)$ 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.