The height process of a continuous state branching process with interaction

TL;DR

This paper constructs the height process for a generalized continuous state branching process with interaction, extending previous models and establishing a connection to population size via an extended Ray--Knight theorem.

Contribution

It introduces a new construction of the height process for processes with interaction and non-vanishing diffusion, broadening the scope of prior models.

Findings

Established the height process for interacting continuous state branching processes.

Extended the second Ray--Knight theorem to include interaction.

Generalized earlier work to processes with non-vanishing diffusion.

Abstract

For a generalized continuous state branching process with non-vanishing diffusion part, finite expectation and a directed ("left-to-right") interaction, we construct the height process of its forest of genealogical trees. The connection between this height process and the population size process is given by an extension of the second Ray--Knight theorem. This paper generalizes earlier work of the two last authors which was restricted to the case of continuous branching mechanisms. Our approach is different from that of Berestycki et al. There the diffusion part of the population process was allowed to vanish, but the class of interactions was more restricted.