From multitype branching Brownian motions to branching Markov additive processes
Abstract
We study a class of multitype branching Lévy processes, where particles move according to type-dependent Lévy 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).
Summary
This paper investigates a class of multitype branching Lévy processes, generalizing multitype branching Brownian motions (BBMs). In these processes, particles move according to type-dependent Lévy processes, switch types via an irreducible Markov chain, and branch according to type-dependent laws. The main research question is to develop a framework for analyzing these processes and to address open problems related to convergence and traveling wave solutions. The authors employ techniques from Markov additive processes (MAPs) and introduce a novel spine decomposition. This spine decomposition allows them to prove convergence results for additive martingales and derivative martingales, and to establish the existence and uniqueness of traveling wave solutions to the corresponding multitype Fisher-Kolmogorov-Petrovskii-Piskounov (FKPP) equations. A key application is resolving open problems posed by Blath et al. (2025) concerning on-off branching Brownian motion models. The significance of this work lies in providing a systematic approach to study branching systems with type-dependent Lévy processes. The spine decomposition provides a powerful tool for analyzing these models. The results on martingale convergence and traveling wave solutions contribute to a deeper understanding of the behavior of multitype branching processes and their connections to FKPP equations. Specifically, the paper addresses the uniqueness of traveling waves and provides a probabilistic representation of them, akin to the Lalley-Sellke construction, for on-off BBMs. The paper also handles the convergence of martingales and existence of traveling waves in the critical regime, which were previously open questions.
Key Insights
- •Novel Spine Decomposition: The authors introduce a new spine decomposition tailored for multitype branching Lévy processes, which is crucial for analyzing the martingale convergence and traveling wave solutions.
- •MAP Framework: The paper leverages the Markov additive process (MAP) framework to provide a clearer view of the underlying structure of the branching system and to develop the necessary mathematical tools to handle the heterogeneity in particle types' dynamics.
- •L1 Convergence Conditions: The paper establishes specific conditions (θλ'(θ) < λ(θ) and finite (k log k) moment for offspring distribution) for the L1-convergence of the additive martingale, which are essential for understanding the asymptotic behavior of the process.
- •Critical Regime Analysis: The paper provides new results on martingale convergence and the existence and uniqueness of traveling wave solutions in the critical regime (λ(θ*) = θ*λ'(θ*)), addressing open questions from previous work.
- •Velocity of Leftmost Particle: Under specific assumptions, the paper derives the velocity of the leftmost particle as -λ(θ*)/θ*, providing insights into the spatial expansion of the branching process.
- •Connection to FKPP Equations: The paper establishes a connection between the branching process and FKPP-type equations, showing that the limits of martingales provide probabilistic representations of traveling wave solutions.
- •Uniqueness of Traveling Waves: The paper proves the uniqueness of traveling wave solutions under the assumption of spectrally negative jumps for the underlying MAP, highlighting a limitation that the authors believe can be relaxed in future work.
Practical Implications
- •Population Dynamics Modeling: The research provides a more realistic framework for modeling population dynamics where individuals can switch between different states (e.g., active/dormant) and exhibit type-dependent movement and reproduction.
- •Algorithm Design for Stochastic Systems: The spine decomposition technique can be adapted for designing efficient simulation algorithms for complex stochastic systems, particularly those involving branching and spatial movement.
- •Understanding Spatial Spread: The results on the velocity of the leftmost particle and the existence of traveling wave solutions are relevant for understanding and predicting the spatial spread of populations or diseases.
- •Applications in Biology and Ecology: The on-off branching Brownian motion model, which is a special case of the authors' framework, has direct applications in modeling dormancy and seed bank effects in biological populations.
- •Future Research Directions: The paper opens up several avenues for future research, including generalizing the model to non-local branching, studying the necessary conditions for the non-triviality of the derivative martingale limit, and analyzing the fine structure of the leftmost position and extremal process.