Pruning a L\'evy continuum random tree

TL;DR

This paper introduces a pruning method for Le9vy continuum random trees using marks and Le9vy snake techniques, demonstrating that the pruned tree remains a Le9vy CRT and analyzing the joint law of excursion lengths.

Contribution

It develops a novel pruning procedure for Le9vy CRTs and proves the pruned tree retains the Le9vy CRT structure, with detailed law characterizations.

Findings

Pruned trees are still Le9vy continuum random trees.

Established a Markov property and martingale framework for exploration processes.

Derived the joint law of excursion lengths before and after pruning.

Abstract

Given a general critical or sub-critical branching mechanism, we define a pruning procedure of the associated L\'evy continuum random tree. This pruning procedure is defined by adding some marks on the tree, using L\'evy snake techniques. We then prove that the resulting sub-tree after pruning is still a L\'evy 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.