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From multitype branching Brownian motions to branching Markov additive processes

Yutao Liang, Yan-Xia Ren, Quan Shi, Fan Yang

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).

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).
Paper Structure (37 sections, 25 theorems, 195 equations)

This paper contains 37 sections, 25 theorems, 195 equations.

Table of Contents

  1. Introduction
  2. A motivating model: on-off branching Brownian motions
  3. Multitype branching Lévy processes viewed as branching MAPs
  4. Main results
  5. Related works
  6. Examples: two-type BBMs
  7. A connection with in BlaEtAl25
  8. A connection with Champneys95
  9. Perspectives and further questions
  10. 1. Generalization to non-local branching with infinite branching rate
  11. 2. Necessary conditions for the non-triviality of the derivative martingale limit
  12. 3. Uniqueness of travelling waves
  13. 4. Finer study of the leftmost position and extremal process
  14. 5. Infinitely many types
  15. Organization of the paper
  16. ...and 22 more sections

Key Result

Proposition 1.2

Let $(\chi(t), \Theta(t))_{t\ge0}$ be a MAP. Let $0=T_0<T_1<\dots$ be successive jump times of $\Theta$. For each $i,j\in \mathcal{I}$, there exist an i.i.d. sequence of random variables $(U_{ij}^{n}, n\ge 1)$, and an i.i.d. sequence of Lévy processes $(\chi_i^{n},n\ge 1)$, such that these $\Theta,\ where $i=\Theta(T_n-)$ and $j = \Theta(T_n)$.

Theorems & Definitions (47)

  • Definition 1.1: Markov additive process (MAP)
  • Proposition 1.2
  • Theorem 1.3: Matrix exponent
  • Theorem 1.4: $\mathcal{L}^1$-convergence
  • Corollary 1.5: Velocity of the leftmost particle
  • Theorem 1.6: Critical derivative martingale
  • Definition 1.7: Travelling waves
  • Theorem 1.8: Existence and uniqueness of travelling waves
  • Proposition 2.1: Iva2011
  • Lemma 2.2
  • ...and 37 more