Crump–Mode–Jagers Processes

• Crump–Mode–Jagers branching processes are a broad framework for modeling continuous-time populations with age-dependent and non-Markovian reproduction events.
• They employ a rigorous mathematical formulation to analyze growth, extinction, explosion, and genealogical structures through concepts like the Malthusian parameter and renewal theory.
• Applications include random tree analysis, diffusion approximations, and scaling limits, connecting classical branching models with modern stochastic processes in biology and network theory.

A Crump–Mode–Jagers (CMJ) branching process is a highly general framework for modeling the stochastic evolution of populations in continuous time, permitting arbitrary dependencies of reproduction events on individual “age,” and allowing birth times to constitute non-Markovian point processes. This formalism encapsulates and extends classical Galton–Watson and Bellman–Harris processes, giving a unified language and set of techniques for age-dependent, inhomogeneous, and spatial branching. The theory supports a robust analysis of population dynamics, extinction, explosion, and the genealogical architecture of random tree-like structures.

1. Formal Construction and Key Properties

A standard CMJ process involves a population structured as a rooted tree or forest, with each individual endowed at birth with a random lifespan and a random reproduction process (usually a point process on the non-negative real line). Let $\mathcal{U}_\infty = \bigcup_{n\ge0}\mathbb N^n$ be the Ulam–Harris tree. Each node $u\in\mathcal{U}_\infty$ is assigned:

• A (possibly random) lifetime $\Lambda_u>0$.
• A point process $\xi_u$ on $[0,\Lambda_u)$ (relative birth-ages of its offspring).

The process is constructed recursively: the root is born at time $0$, lives for $\Lambda_\varnothing$, and bears children at times $\{\tau_1, \tau_2, \ldots\} = \xi_\varnothing$. The $i$th child of the root, denoted $u_i$, is born at time $\tau_i$ and thereafter undergoes an independent copy of the same birth process. The total population at time $t$ is

$$Z(t) = \sum_{u\in\mathcal{U}_\infty} \mathbf{1} \left\{ \mathcal{B}(u) \le t < \mathcal{B}(u) + \Lambda_u \right\}$$

where $\mathcal{B}(u)$ is the birth time of node $u$, given by the cumulative sum of birth intervals along the path from root to $u$.

A non-negative characteristic $\phi:[0,\infty)\rightarrow \mathbb{R}_+$ can be used to count general functionals:

$$Z^\phi(t) = \sum_{u:\ \mathcal{B}(u) \le t} \phi(t - \mathcal{B}(u))$$

The process $(Z(t))_{t\ge0}$ thus constructed is not, in general, Markovian unless further structure (e.g., exponential lifetimes and Poissonian births) is imposed. However, it is regenerative and accommodates the full spectrum of age dependence.

2. Malthusian Parameter and Population Growth Asymptotics

The mean total offspring measure of a single individual is $\mu(dt) = \mathbb{E}[\xi(dt)]$. Provided that $\mu$ is nonlattice and $\mu(\{0\})<1$, the Malthusian parameter $\alpha>0$ is defined as the unique solution to

$\int_0^\infty e^{-\alpha t}\mu(dt) = 1$

Under supercriticality ($\mathbb E[\xi([0,\infty))] > 1$), the classical Jagers–Nerman theorem states that the normalized process $W(t) = e^{-\alpha t}Z(t)$ converges a.s. and in $L^1$ to a nontrivial limit $W(\infty)$ on the event of nonextinction, provided a moment condition $$\mathbb E [\xi([0,\infty))\log^+ \xi([0,\infty))] < \infty$$ is satisfied (Holmgren et al., 2016).

For general characteristics,

$$e^{-\alpha t}Z^\phi(t) \xrightarrow{a.s.} W(\infty) \frac{1}{\alpha}\int_0^\infty e^{-\alpha s}\mathbb E[\phi(s)]ds$$

These results enable the analysis of not just population size, but also broad statistics and patterns within the genealogical tree, such as subtree frequencies and structural properties (Plazzotta et al., 2015).

3. Scaling Limits, Diffusion Approximations, and Functional CLT

In various regimes, CMJ processes admit diffusion-type scaling limits:

• For binary homogeneous/recurrent CMJ processes, under appropriate rescaling (high birth rates, or large populations with forward-recurrence lifetime initialization), convergence to continuous-state branching diffusions (Feller diffusion) or reflected Brownian motion is established (Lambert et al., 2011, Xu, 2018). Explicitly, starting from $n$ ancestors and letting $n\to \infty$,

$Z_n(t) = \frac{1}{n} z_n(n t) \implies Y(t)$

where $Y$ is a Feller diffusion satisfying

$$dY_t = -\frac{\alpha}{\beta}Y_t dt + \sqrt{\frac{1}{\beta}Y_t}dB_t$$

with parameters determined by the underlying lifetime and birth process (Lambert et al., 2011).

• With further generality (immigration, randomness), weak convergence results hold for CMJ with immigration to continuous-state branching processes with immigration (CBI), as solutions to certain SDEs with jumps, in the sense of Mitoma (Xu, 2018).
• Central limit theorems for supercritical binary homogeneous CMJ have been established, showing that after exponential normalization, the error converges in distribution to a Laplace law (mixture of Gaussians) (Henry, 2015).
• Functional law of large numbers and central limit theorems are available for age-dependent, random birth-rate CMJ processes. If $N$ initial ancestors, then

$\frac{1}{N} Z^{N,x}(t) \to x M_1(t)$

with Gaussian fluctuations described by a process with explicit covariance determined by the spread in lifetimes, birth-rates, and Poisson noise (Dramé et al., 16 Aug 2025).

4. Criteria for Explosion and Pathologies

A key phenomenon in the theory is “explosion”: infinitely many individuals are born in finite time. For a general CMJ process, explosion is governed by the fine structure of the reproduction process:

• Fixed-Point and Smoothing Transform: The probability of non-explosion $F(t)$ solves the fixed-point equation

$$F(t) = \mathbb{E} \left[ \prod_{i=1}^N F(t - S_i) \right]$$

where $N$ is the number of immediate children and $S_i$ their birth times (Alsmeyer et al., 7 Mar 2025).

• Explicit explosion criteria: For pure birth CMJ processes with birth rates $(\lambda_i)$,
  • $\sum 1/\lambda_i < \infty$ suffices for explosion.
  • Suitable lower bounds on local birth rate growth (e.g., $\lambda_i/i \to \infty$ with specified sum conditions) ensure non-explosion.
  • Sharpness: There exist rate sequences with $\sum 1/\lambda_i = \infty$ but still explosive due to block construction with extreme oscillations (Galganov et al., 11 Jan 2026).
• Coupling/Comparison Techniques: Explosion is preserved under partial couplings that shift births earlier or increase offspring-tail heaviness (Komjathy, 2016). Min-summability or "thinning" constructions yield tractable integral criteria for explosion in age-dependent branching.
• Exotic explosion: There exist birth-time distributions that are singular wrt Lebesgue, with arbitrarily "exotic" structure, yet still yield explosive processes under suitable offspring tail behavior (Komjathy, 2016).
• Applications to network trees: In random tree models such as recursive trees or preferential attachment with fitness, explosion in the corresponding CMJ process manifests as the presence of a vertex of infinite degree (infinite star) or an infinite path, with precise phase diagrams connecting parameters controlling birth rates and node "fitness" to condensation or backbone regimes (Iyer, 2023, Lodewijks, 2024, Iyer et al., 2023).

5. Tree Shapes, Structural Asymptotics, and Moment Analysis

Crump–Mode–Jagers processes provide a framework for detailed analysis of the genealogy of the random tree:

• Shape Frequency: Using random characteristics, the asymptotic relative frequency (per leaf) of any subtree shape (e.g., cherries, pitchforks) in supercritical CMJ trees is computable:

$$F_{\mathcal{S}} = M \int_0^\infty e^{-M t} \mathbb{E}[\phi^{\mathcal{S}}(t)]dt$$

where $M$ is the Malthusian parameter and $\phi^{\mathcal{S}}$ the indicator for shape $\mathcal{S}$ (Plazzotta et al., 2015).

• Martingale and Renewal Methods: Exponential normalization yields martingales that converge almost surely and in $L^p$ under appropriate conditions (Holmgren et al., 2016, Móri et al., 2018). Renewal equations for moments at all orders provide sharp $L_k$-boundedness results for general time-dependent characteristics.
• Applications: These methods yield detailed results for node degree distributions, subtree counts, height, path length, expected number of protected nodes, maximal clades in m-ary search trees, and the fine profile of the entire genealogy (Holmgren et al., 2016).

6. Extensions: Interaction, Nonlinear Effects, and Random Environments

Recent work explores far-reaching generalizations of the CMJ paradigm:

• Nonlinear/Interactive CMJ: Processes with interaction—e.g., individuals give birth at rate dependent on the current age-structure mesured by an empirical measure—can be analyzed via propagation of chaos and convergence of empirical measures in the local topology to deterministic profiles governed by nonlinear renewal PDEs (Foutel-Rodier et al., 13 Nov 2025).
• Genealogy in Random Environment: Crump–Mode–Jagers models indexed by node fitness, random birth rates, or more general random environments, permit both variable exponential and subexponential growth, and admit phase transitions between condensation, existence of an infinite star, or an infinite path backbone depending on precise tail behavior of the fitness and birth law (Lodewijks, 2024, Iyer et al., 2023).
• Self-Similar and Branching-Stable Measures: The branching-stable point measures of negative scaling exponent, constructed as static immortal CMJ processes with scale-free reproduction kernels, yield scaling relations, detailed asymptotics for cumulative counts, and explicit martingale convergence theorems following the paradigm established by Biggins for branching random walks (Bertoin et al., 2018).

7. Excursion Theory, Path Decompositions, and Connections to Lévy Processes

Excursion-theoretic techniques offer powerful representations and invariance principles:

• Lévy Excursion Encoding: The contour process of a splitting tree with lifespan measure $\Pi$ is a reflected spectrally positive Lévy process, and its local time at height $a$ counts the alive population at age $a$ (Felipe et al., 2016, Lambert et al., 2011).
• Space-Time Reversal and Williams’ Decomposition: For (sub)critical CMJ, the genealogy viewed backward from extinction is equal in law to the forward process; pre- and post-supremum segments of excursions are invariant under space-time reversal (Felipe et al., 2016).
• Scaling Connections with Queues: The Lamperti transformation, excursion theory, and reflected Brownian limits demonstrate an equivalence between rescaled CMJ genealogies and heavy-traffic processor-sharing queue dynamics (Lambert et al., 2011).
• Measure-Valued Particle Systems: The total mass of branching particle systems coded by Lévy excursions is precisely a CMJ process, with measure-valued Markov structure and renewal equations for moments (He et al., 2012).

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The Crump–Mode–Jagers formalism thus underpins a vast corpus of population models, random graph dynamics, and sophisticated stochastic genealogies, with a wealth of limit theorems, structural characterizations, and deep correspondences to classical processes in probability theory. The rich mathematical structure enables precise analysis of population growth, extinction and explosion, fluctuations, limiting genealogies, and the emergence of complex phenomena such as condensation or backbone formation, providing tools and insights central to modern mathematical biology, random networks, and branching systems (Holmgren et al., 2016, Xu, 2018, Komjathy, 2016, Iyer, 2023, Iyer et al., 2023, Foutel-Rodier et al., 13 Nov 2025, Lambert et al., 2011, Felipe et al., 2016).