Multitype Branching Lévy Processes

• Multitype Branching Lévy Processes are stochastic models that incorporate type-specific Lévy motions, Markov chain switching, and reproduction laws to capture population evolution.
• The framework unifies spatial movement, genealogical structure, and extinction criteria using analytical tools like spine decomposition and additive martingales.
• Scaling limits link discrete models to continuous Lévy trees and non-linear FKPP equations, revealing detailed ancestral and mutation structures.

A multitype branching Lévy process is a stochastic population model in which individuals of various types move according to type-specific Lévy processes, switch types via a Markov chain, and branch according to type-dependent reproduction laws. This framework naturally generalizes both the classic branching Brownian motion and continuous-state multitype branching processes, embedding spatial movement, genealogical structure, and type interactions in a unified Markov additive process (MAP) perspective. The theory includes both discrete and continuum genealogies, coalescent structures, extinction criteria, scaling limits, and connections with nonlinear PDEs of the FKPP type.

1. Mathematical Formulation and Model Structure

Consider a type space $\mathcal{I}=\{1,\dots, K\}$. Each particle in the process is characterized by its spatial position $x\in\mathbb{R}$ and its type $i\in\mathcal{I}$. The evolution is as follows:

• Motion: For each $i\in\mathcal{I}$, particles of type $i$ move according to a Lévy process $\chi^i(t)$ with Laplace exponent $$\phi_i(\theta) = \log \mathbb{E}[e^{-\theta \chi^i(1)}]$$.
• Type switching: Each particle's type evolves as an irreducible continuous-time Markov chain $\Theta(t)$ with generator matrix $Q=(q_{ij})$. Upon switching $i\to j$, the particle may jump spatially by an amount distributed as $U_{ij}$ with Laplace transform $G_{ij}(\theta)=\mathbb{E}[e^{-\theta U_{ij}}]$.
• Branching: While at type $i$, a particle branches at rate $\beta_i$ into a random number of offspring (possibly zero), distributed according to the law $\mu_i$, all at the same location and type.

These mechanisms collectively lead to a population whose empirical field is a Markov branching process on $\mathbb{R}\times\mathcal{I}$, where the (joint) generator for test functions $f\in C^2(\mathbb{R}\times\mathcal{I})$ is

$$A f(x,i) = A_i f(\cdot,i)(x) + \sum_{j\neq i} q_{ij} \int [f(x+y,j) - f(x,i)] P(U_{ij}\in dy),$$

with $A_i$ the Lévy generator for type $i$ and $U_{ij}$ as above (Liang et al., 24 Dec 2025).

2. Spine Decomposition and Martingale Techniques

A central analytical tool is the spine decomposition via a many-to-one change of measure using additive martingales,

$$W_\theta(t) = \sum_{u\in N_t} e^{-\theta X_u(t) - \lambda(\theta) t} V_{J_u(t)}(\theta),$$

where $X_u(t)$ is the position of particle $u$, $J_u(t)$ its type, $V(\theta)$ a right Perron-Frobenius eigenvector, and $\lambda(\theta)$ the corresponding eigenvalue of a "first-moment" matrix intertwining spatial motion, type switching, and branching. This additive martingale permits a spine change of measure, under which the law of a randomly chosen lineage (the spine) is tilted in space and type, and branching events become size-biased (Liang et al., 24 Dec 2025).

Key results:

• $W_\theta(t)$ is a positive martingale, converging in $L^1$ under integrability conditions (e.g., $\theta \lambda'(\theta)<\lambda(\theta)$ and $\sum_k k (\ln k) \mu_j(k)<\infty$ for all $j$).
• The minimal position $L_t = \min_{u\in N_t} X_u(t)$ obeys a law of large numbers, $L_t/t \to -\lambda(\theta^*)/\theta^*$ for $\theta^*$ solving $\lambda(\theta^*) = \theta^* \lambda'(\theta^*)$.
• At criticality, a derivative martingale $Z(t)$ converges to a non-degenerate limit under stronger integrability (Liang et al., 24 Dec 2025).

3. Scaling Limits: Multitype Lévy Trees and Fields

Scaling limits of discrete multitype population models (such as Bienaymé-Galton-Watson trees with type-dependent offspring laws) yield "multitype Lévy trees". In this regime:

• Each type $i$ is encoded via a spectrally positive Lévy excursion (with possible cross-type immigration), and the joint evolution is described by a $d\times d$ matrix of Lévy fields, each component $X^{i,j}$ a Lévy process—spectrally positive for $i=j$ and a subordinator for $i\neq j$ (Hernández et al., 6 Feb 2025).
• The full multitype Lévy tree is realized by gluing these single-type components according to the Poissonian structure of cross-type jumps (the "decorations") using Gromov–Hausdorff–Prohorov topology. Vector-valued measure-marked spaces track genealogies and type composition at all scales (Hernández et al., 6 Feb 2025).

The main invariance principle states that, after suitable rescaling and conditioning on large subtrees, sequences of multitype BGW trees converge in law to these Lévy trees, with joint genealogy governed by the limiting spectrally positive additive Lévy field (Hernández et al., 6 Feb 2025).

4. Genealogy, Coalescence, and Ancestral Structure

For multitype branching processes (especially CSBPs), Poissonization enables analysis of the coalescent clock for random samples from the living population:

• A uniform $k$-sample from a multitype population at time $T$ is distributionally equivalent to a mixture of independent Poisson samples with random rates, with the mixing determined by the structure of the population and the "forest" of ancestral lines (Johnston et al., 2019).
• The full ancestral forest for samples from all types is explicitly described via an energy functional involving derivatives of the Laplace exponent.
• In the small time limit, the genealogy of a small sample converges to a multitype $\Lambda$-coalescent, generalizing Kingman's and Bolthausen–Sznitman coalescents to the multitype, continuous-state setting.
• Special cases (e.g., multidimensional Feller diffusion) admit explicit coalescence rates: pairwise coalescence within a type at rate $2B_i/x_i$, inter-type "type-switch" at rate $K_{c,j}/x_j$ (Johnston et al., 2019).

5. Extinction Criteria and Lamperti Representation

Multitype CSBPs and their extinction properties are governed by the vector-valued branching mechanism $\psi=(\psi_1,\dots,\psi_d)$, encoding type-specific and inter-type reproduction. Pathwise, the multitype Lamperti transformation represents the population as the time-change of independent spectrally positive $\mathbb{R}^d$-valued Lévy processes (Chaumont et al., 2021).

• The extinction probability is given by

$$\mathbb{P}_r(\lim_{t\to\infty} Z_t=0) = \exp(-\langle r, \phi(0) \rangle),$$

where $\phi$ is the (componentwise) inverse of the branching mechanism $\psi$ (Chaumont et al., 2021).

• Finite-time extinction is decided by a multitype extension of Grey's integral test: For each type $i$, let $\psi_i(s) = \psi_i(se_i)$. Let $G$: $\int^\infty ds / \psi_i(s)<\infty$ for every $i$. If (G) holds for all $i$, extinction occurs in finite time with probability $\exp(-\langle r, \phi(0)\rangle)$. If (G) fails for some $i$, extinction (if it occurs) is only in the limit $t\to\infty$.
• Proofs use fluctuation theory for spectrally positive additive Lévy fields, analysis of multivariate first-hitting times, and comparison ODE arguments (Chaumont et al., 2021).

6. FKPP Equations and Travelling Waves

The population's spatial and genealogical evolution connects to nonlinear PDEs. For the process above, the multitype FKPP system is

$$\partial_t u_i = A_i u_i + \sum_{j\neq i} q_{ij} (\int u_j(t,x+y) P(U_{ij}\in dy) - u_i(t,x)) + \beta_i(g_i(u_i) - u_i),$$

where $g_i(s) = \sum_k \mu_i(k) s^k$. Existence and uniqueness of traveling wave solutions correspond to convergence of additive and derivative martingales, and are established via spine and genealogical decompositions. For example, for traveling wave speed $c$, existence holds if and only if $c \geq \lambda(\theta^*)/\theta^*$ (Liang et al., 24 Dec 2025).

7. Scaling Limits, Marked Jumps, and Mutation Structures

Branching Lévy processes with marked (mutation) jumps encode both genealogy and mutation history. The scaling limit of such population models leads to Poisson point processes governing the depths (ladder heights) of excursions, and the mutations' spatial-temporal distribution can be recovered from the marked ladder-height process. In the critical branching case with exponential lifetimes, this yields explicit formulae for the genealogical and mutational partition (e.g., number of mutations along a lineage is Poisson with mean proportional to coalescence depth) (Delaporte, 2013).

• Y. Liang, Y.–X. Ren, Q. Shi, F. Yang. "From multitype branching Brownian motions to branching Markov additive processes" (Liang et al., 24 Dec 2025).
• Johnston & Lambert. "The coalescent structure of uniform and Poisson samples from multitype branching processes" (Johnston et al., 2019).
• Chaumont & Marolleau. "Extinction times of multitype, continuous-state branching processes" (Chaumont et al., 2021).
• Abraham & Delmas. "Multitype Lévy trees as scaling limits of multitype Bienaymé-Galton-Watson trees" (Hernández et al., 6 Feb 2025).
• Delmas, Siri-Jégousse. "Lévy processes with marked jumps II: Application to a population model with mutations at birth" (Delaporte, 2013).