From multitype branching Brownian motions to branching Markov additive processes

TL;DR

This paper introduces a generalized framework for multitype branching processes with type-dependent Lévy motions and Markovian type switching, extending previous models and solving open problems in the field.

Contribution

It develops a spine decomposition technique for multitype Lévy processes and proves convergence of martingales and existence of traveling wave solutions, generalizing multitype branching Brownian motions.

Findings

Proves convergence of additive and derivative martingales.

Establishes existence and uniqueness of traveling wave solutions.

Resolves open problems in on-off branching Brownian motion models.

Abstract

We study a class of multitype branching L\'evy processes, where particles move according to type-dependent L\'evy processes, switch types via an irreducible Markov chain, and branch according to type-dependent laws. This framework generalizes multitype branching Brownian motions. Using techniques of Markov additive processes, we develop a spine decomposition. This approach further enables us to prove convergence results for the additive martingales and derivative martingales, and establish the existence and uniqueness of travelling wave solutions to the corresponding multitype FKPP equations. In particular, applying our results to the on-off branching Brownian motion model resolves several open problems posed by Blath et al.(2025).