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

Abstract

By studying an admissible family of branching mechanisms introduced in Li (2014), we obtain a pruning procedure on Lévy trees. Then we could construct a decreasing Lévy-CRT-valued process $\{{\mathcal{T}}_t\}$ by pruning Lévy trees and an analogous process $\{{\mathcal{T}}_t^{\ast }\}$ by pruning a critical Lévy 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 $\{{\mathcal{T}}_t^{\ast }\}$. The results generalize those studied in Abraham and Delmas (2012).

25 pages. Version 2 revised the title and improved exposition in various places

References in corpus (4)

• Random trees and applications
• A continuum-tree-valued Markov process
• Growth of Levy trees
• Exit times for an increasing Lévy tree-valued process

Cited by in corpus (1)

• Invariance principles for pruning processes of Galton-Watson trees