Spinal constructions for continuous type-space branching processes with interactions

TL;DR

This paper develops a new mathematical framework using spinal constructions to analyze continuous-time structured branching processes with interactions, enabling better understanding and simulation of complex population dynamics.

Contribution

It introduces a Girsanov-type result and a generalized Kesten-Stigum theorem for interacting populations, along with an alternative simulation method based on spine construction.

Findings

Derived a many-to-one formula for interacting populations

Established a generalized Kesten-Stigum theorem with interactions

Proposed an exact simulation approach for size-dependent populations

Abstract

We consider branching processes describing structured, interacting populations in continuous time. Dynamics of each individuals characteristics and branching properties can be influenced by the entire population. We propose a Girsanov-type result based on a spinal construction, and establish a many-to-one formula. By combining this result with the spinal decomposition, we derive a generalized continuous-time version of the Kesten-Stigum theorem that incorporates interactions. Additionally, we propose an alternative approach of the spine construction for exact simulations of stochastic size-dependent populations.