Mass erasure on measured $\mathbb{R}$-trees, applications to Lévy forests

Abstract

Let $h>0$. For a complete and separable $\mathbb{R}$-tree $(T,d)$ equipped with a root $ρ$ and a finite Borel measure $μ$, we define the $h$-mass-erased tree by removing from $T$ all fringe subtrees of mass less than $h$ and we equip it with a suitable measure such that the erasure operators $(\mathcal{E}_h)_{h\ge 0}$ form a semigroup that is continuous for the Gromov-weak topology. Then, we say that a sequence $\boldsymbolμ_n=(T_n,d_n,ρ_n,μ_n)$, $n\in\mathbb{N}$, converges in the sense of mass erasure if $(\mathcal{E}_h\boldsymbolμ_n)_{n\in\mathbb{N}}$ converges Gromov-weakly for all $h\ge 0$. This notion of convergence is strictly weaker than Gromov-weak convergence and we establish criteria to relate the two notions. We define a distance function that metrizes convergence in the sense of mass erasure. By extending the notion of measured $\mathbb{R}$-trees to allow mass on the boundary (the far ends of infinite geodesics), we obtain a complete metric space. Next, we identify random trees of finite type (that is, discrete trees with edge lengths) satisfying the regenerative branching property as a specific class of measured (sub)critical GW forests. We then show that this class of trees is preserved by mass erasure and we compute the law of these mass-erased GW forests explicitly. Finally, we establish a limit theorem for these measured (sub)critical GW forests to converge to standard measured Lévy forests, i.e. those whose total mass has the same distribution as the total population of a continuous-state branching process. This includes cases with bounded variation by crucially using the convergence in the sense of mass erasure and it extends the cases studied previously.