Ray-Knight representation of flows of branching processes with competition by pruning of L\'evy trees

TL;DR

This paper introduces a new class of flows for continuous state branching processes with competition, using stochastic differential equations and a novel genealogical pruning approach on Lévy trees, with a Ray-Knight representation.

Contribution

It develops a new genealogical description of branching processes with competition via interactive pruning of Lévy trees and establishes a Ray-Knight representation for these processes.

Findings

Constructed processes as solutions to stochastic differential equations.

Proposed a genealogical description based on pruning Lévy trees.

Established a Ray-Knight representation linking local times and pruned forests.

Abstract

We introduce flows of branching processes with competition, which describe the evolution of general continuous state branching populations in which interactions between individuals give rise to a negative density dependence term. This generalizes the logistic branching processes studied by Lambert. Following the approach developed by Dawson and Li, we first construct such processes as the solutions of certain flow of stochastic differential equations. We then propose a novel genealogical description for branching processes with competition based on interactive pruning of L\'evy-trees, and establish a Ray-Knight representation result for these processes in terms of the local times of suitably pruned forests.