Pruning a Lévy Continuum Random Tree

Abstract

Given a general critical or sub-critical branching mechanism, we define a pruning procedure of the associated Lévy continuum random tree. This pruning procedure is defined by adding some marks on the tree, using Lévy snake techniques. We then prove that the resulting sub-tree after pruning is still a Lévy continuum random tree. This last result is proved using the exploration process that codes the CRT, a special Markov property and martingale problems for exploration processes. We finally give the joint law under the excursion measure of the lengths of the excursions of the initial exploration process and the pruned one.

A Construction of a β-Coalescent via the Pruning of Binary Trees

“…A coalescent process may also be viewed as a process (I(t), t ≥ 0) taking values in the interval partitions of [0,1]. The length of each interval represents the mass of a block, and the process represents blocks (whose sizes sum to 1) that merge together as time passes.…”

“…A pruning theory of such a continuum tree has been introduced in [5] (see [4] for a general theory of the pruning of Lévy trees) and will be recalled in Section 3.1. Using this pruning procedure, we are able to define an interval-partition-valued process in Section 3.2 which has the same structure as the β(3/2, 1/2)-coalescent except for the times (when sampling n points uniformly distributed on (0, 1), the time interval between two coalescences is not exponentially distributed).…”

“…As in [5,1], we throw points on the CRT "uniformly" on the skeleton of the CRT and add more and more points as time goes by. More precisely, we consider a Poisson point measure M (dθ, dx) on [0, +∞) × T with intensity 4 dθℓ(dx).…”

“…Let Λ be a finite measure on [0,1]. A Λ-coalescent (Π(t), t ≥ 0) is a Markov process which takes values in the set of partitions of N * = {1, 2, .…”

“…This work is partially supported by the "Agence Nationale de la Recherche", ANR-08-BLAN-0190. 1 an exponential time of parameter: (3) λ n = (n − 1) β 1 2 , n − 1 2 , choose one of the n − 1 inner vertices of the tree uniformly at random, coalesce all the leaves of the subtree attached at the chosen node and remove that subtree from the original tree.…”