A tree-valued Markov processes associated with an admissible family of branching mechanisms

TL;DR

This paper constructs and analyzes tree-valued Markov processes derived from pruning Lévy trees, extending previous work by exploring their laws at specific times and under certain conditions.

Contribution

It introduces a new pruning procedure for Lévy trees based on an admissible family of branching mechanisms, generalizing earlier results.

Findings

Constructed decreasing Lévy-CRT-valued processes via pruning.

Represented the law of the process at the ascension time.

Extended previous models to a broader class of branching mechanisms.

Abstract

By studying an admissible family of branching mechanisms introduced in Li (2014), we obtain a pruning procedure on L\'evy trees. Then we could construct a decreasing L\'evy-CRT-valued process $\{{\mathcal T}_t\}$ by pruning L\'evy trees and an analogous process $\{{\mathcal T}^*_t\}$ by pruning a critical L\'evy tree conditioned to be infinite. Under a regular condition on the admissible family of branching mechanisms, we show that the law of $\{{\mathcal T}_t\}$ at the ascension time can be represented by $\{{\mathcal T}^*_t\}$. The results generalize those studied in Abraham and Delmas (2012).