Linear and uniform in time bound for the binary branching model with Moran type interactions

TL;DR

This paper reviews the binary branching model with Moran interactions, relating it to the Feynman-Kac semigroup, and improves bounds on their L2 distance under certain regularity conditions.

Contribution

It provides a clearer understanding of the BBMMI model and establishes improved bounds on the L2 distance between related normalizations with regularity assumptions.

Findings

Improved L2 distance bounds for the model

Enhanced understanding of the relation to Feynman-Kac semigroup

Clarification of the model's dynamics under regularity conditions

Abstract

In this note, we recall the definition of the binary branching model with Moran type interactions (BBMMI) introduced in [8]. In this interacting particle system, particles evolve, reproduce and die independently and, with a probability that may depend on the configuration of the whole system, the death of a particle may trigger the reproduction of another particle, while a branching event may trigger the death of another particle. We recall its relation to the Feynman-Kac semigroup of the underlying Markov evolution and improve on the L 2 distance between their normalisations proved in [8], when additional regularity is assumed on the process.