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 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).