Continuous-State Branching Processes

• Continuous-state branching processes are Markov models defined on [0, ∞) using Lévy–Khintchine type branching mechanisms to capture the evolution of continuous populations.
• They serve as scaling limits of discrete branching models, with explicit criteria for extinction and explosion and extensions incorporating immigration and random environments.
• Analytic tools such as the Lamperti transform, Dirichlet forms, and scale functions provide robust frameworks for studying genealogical structures and long-term behavior.

Continuous-state branching processes (CSBPs) are stochastic processes that model the evolution of a continuous @@@@1@@@@ or population, governed by branching mechanisms generalizing the classical discrete-state Galton–Watson processes. They serve as scaling limits of discrete branching models and are described via canonical stochastic equations, semigroups, and probabilistic constructions that encode intricate behaviors such as extinction, explosion, path decomposition, interactions with immigration, environmental variability, collisions, self-regulation, and genealogical structure.

1. Fundamental Characterization and Stochastic Construction

A CSBP $(X_t)_{t \ge 0}$ is a Markov process taking values in $[0, \infty)$, characterized by the branching property: for $x,y \ge 0$, the law of $X_t$ started from $x+y$ is the convolution of independent copies started from $x$ and $y$. The law is determined by a branching mechanism $\Psi$ of Lévy–Khintchine form,

$$\Psi(q) = bq + \sigma^2 q^2 + \int_{(0,\infty)} (e^{-qz} - 1 + qz\mathbb{1}_{\{z \le 1\}}) \pi(dz),$$

where $b \in \mathbb{R}$, $\sigma \geq 0$, and $\pi$ is a Lévy measure. The transition function satisfies

$$\mathbb{E}_x\left[e^{-\lambda X_t}\right] = \exp\left\{ -x v_t(\lambda) \right\},$$

where $v_t(\lambda)$ solves

$$\frac{\mathrm{d}}{\mathrm{d}t} v_t(\lambda) = -\Psi\left( v_t(\lambda) \right), \quad v_0(\lambda) = \lambda.$$

Stochastic integral representations describe $X_t$ as the pathwise unique solution to equations driven by Brownian motion and Poisson random measures (Li, 2012, Li, 2016). For example, a typical SDE for a CSBP is

$$X_t = X_0 + \int_0^t \sqrt{2c X_s} \, dB_s - b \int_0^t X_s \, ds + \int_0^t \int_{(0,1]} \int_0^{X_{s-}} z\,\widetilde{M}(ds, dz, du) + \int_0^t \int_{(1,\infty)} \int_0^{X_{s-}} z\,M(ds, dz, du),$$

where the integrals encode the diffusion and branching jump structure.

2. Scaling Limits, Extinction, and Explosion

CSBPs appear as scaling limits of Galton–Watson branching processes under proper space-time rescaling and “uniform Lipschitz” conditions on offspring generating functions (Li, 2012). Scaling limits ensure the convergence of rescaled Markov chains on increasingly dense subsets of $\mathbb{R}_+$ to the CSBP in $D([0,\infty), \mathbb{R}_+)$.

Extinction probabilities are given by

$P_x\{ T_0 < \infty\} = \exp\{ -x v\},$

where $v = \lim_{t\to\infty} v_t(1)$. Extinction in finite time occurs almost surely if and only if $\int^\infty \frac{du}{\Psi(u)} < \infty$ and $\Psi'(0^+) \geq 0$ (Pardo et al., 2013). Explosion (blow-up in finite time) can also occur, with explicit criteria in terms of the tail behavior of $\Psi$ and corresponding ODEs arising from the generator (Li, 2016, Murillo-Salas et al., 2015). The process can “come down from infinity” under certain regimes, meaning that starting from an infinite mass the process reaches finite values almost surely in finite time (Li, 2016).

A law of the iterated logarithm describes near-extinction fluctuations, and critical or subcritical cases exhibit Khintchin-type LIL behavior for both the process and its trajectory reflected at its running infimum (Pardo et al., 2013).

3. Extensions: Immigration, Random Environments, and Collisions

3.1 Immigration: CBI Processes

The continuous-state branching process with immigration (CBI) covers scenarios where independent “immigrant” mass flows into the system, determined via an immigration mechanism $\Phi(\lambda)$ of Lévy–Khintchine form: $$\Phi(\lambda) = \beta \lambda + \int_{(0, \infty)} (1 - e^{-\lambda u}) \nu(du).$$ The Laplace transform is then: $$\mathbb{E}_x[e^{-\lambda Y_t}] = \exp\left\{ -x v_t(\lambda) - \int_0^t \Phi\big(v_s(\lambda)\big) ds \right\}.$$ Long-time behavior involves two regimes (Foucart et al., 2020, Foucart et al., 2023):

• Branching-dominated regime: $\int_0 \frac{\Phi(u)}{|\Psi(u)|} du < \infty$, linear renormalizations yield almost-sure limits.
• Immigration-dominated/extremal regime: $\int_0 \frac{\Phi(u)}{|\Psi(u)|} du = \infty$; only nonlinear, time-dependent renormalizations can yield nondegenerate weak limits, and extremal or extremal shot noise processes may emerge.

3.2 Environmental Effects

CSBPs in random or varying environments incorporate multiplicative fluctuations driven by external Lévy processes or time-dependent coefficients (Palau et al., 2015, He et al., 2016, Fang et al., 2020, Li et al., 5 Feb 2025). Stochastic integral equations then contain terms like $\int_0^t X_{s-} dL(s)$, or environment-dependent parameters $b(t)$, $c(t)$, and inhomogeneous jump measures. Extinction and explosion criteria and stationary (ergodic) behavior are determined through a combination of branching, immigration, and environmental drift.

3.3 Polynomial and Multitype Models

Polynomials in the state space or the inclusion of multiple types (vector-valued processes) allow for state-dependent branching rates and type interactions (Li, 2016, Li et al., 5 Feb 2025, Li et al., 6 Feb 2025). Pathwise uniqueness and comparison principles, as well as explicit Laplace transform formulas for functionals, generalize classical results to high-dimensional and polynomial-transition settings.

3.4 Collisions and Interaction

Continuous-state branching processes with collisions (CBCs) extend the classical framework by incorporating pairwise interactions, each governed by a (possibly subcritical) Lévy–Khintchine mechanism $\Sigma$ (Foucart et al., 2022). The generator takes the form

$$\mathcal{L}f(z) = z^2 L^\Sigma f(z) + z L^\Psi f(z),$$

entailing boundary classifications and Laplace duality with diffusions. CBCs are tied to continuous-state branching models with migration or logistic-type self-regulation.

4. Analytic Structure: Dirichlet Forms, Invariant Laws, and Sector Constants

The generator and associated (possibly non-symmetric) Dirichlet forms $\mathcal{E}(f,g)$ encode crucial qualitative properties. Stationary distributions are often generalized gamma convolutions (GGCs), uniquely specified via their Laplace exponents and Thorin measures (Handa, 2011). Gamma distributions are the reversible stationary measures; non-reversible stationary laws correspond to non-Gamma GGCs, and the sector constant quantifies the degree of non-symmetry: $|\mathcal{E}(f, g)|^2 \leq C\,\mathcal{E}(f, f)\mathcal{E}(g, g) \quad \text{(symmetry: %%%%27%%%%)}.$ The strong sector condition and associated bounds in terms of the Thorin measure provide analytic control over asymmetry and mixing.

Stationary and quasi-stationary measures (Yaglom distributions) appear under subcritical and critical conditions, with explicit representations via scale functions $W(x)$ satisfying $$\int_0^\infty e^{-\lambda x} W(x) dx = 1/\psi(\lambda)$$ (Liu et al., 2023). Conditional limits and size-biased distributions arise when conditioning on extinction at a fixed time or in the near future.

5. Transformations and Alternative Representations

Several deep correspondences and analytical tools are central in CSBP theory:

• Lamperti Transform: Establishes a one-to-one correspondence between CSBPs and time-changed spectrally positive Lévy processes, with the time change $T(t)=\inf\{ u \geq 0 : \int_0^u X_s ds > t \}$ (Li, 2012, Murillo-Salas et al., 2015).
• Stieltjes Transforms and Noncommutative Probability: Stieltjes transforms associated to Thorin measures provide analytic representations of stationary laws. Boolean convolution and fixed-point type equations link CBI processes to concepts in free probability (Handa, 2011).
• Time-fractional and Non-Markovian Models: Memory effects introduced by random time-changes via inverse stable subordinators yield time-fractional equations and lead to processes with Mittag-Leffler function moments, fractional Fokker–Planck equations, and branching inequalities (Andreis et al., 2017).

6. Genealogy, Flows, and Coalescent Structures

The genealogical structure of CSBPs is revealed through construction as nested flows of subordinators, whose inverses represent the ancestral lineages of current individuals. The inversion (Siegmund duality)

$$\mathbb{P}(\hat{X}_t(y) < x) = \mathbb{P}(X_t(x) > y)$$

describes the dual process with negative jumps corresponding to coalescences (Foucart et al., 2018). Consecutive coalescent processes and the connection to Markovian block mergers (unlike classical exchangeable coalescents) emerge precisely via Poisson-sampled genealogical structures.

The long-term behavior of flows of CSBPs in infinite mean or variation regimes is governed by extremal processes, whose jumps ("super-individuals") dominate population dynamics and connect with maximal processes over Poisson point processes (Foucart et al., 2016). These findings are closely related to the "Eve" property and scaling limits, with applications to genetic diversity and epidemic modeling.

In the context of stationary CSBPs with immigration, genealogical trees are encoded as Lévy trees, and the ancestry of extant populations can be mapped—up to deterministic time change—to continuous-time Galton–Watson processes with immigration, or, in discrete limit, to coalescents such as the Bolthausen–Sznitman coalescent (Abraham et al., 2020).

7. Applications, Special Regimes, and Further Extensions

Continuous-state branching frameworks cover a vast range of applied models:

• Population Biology and Genetics: Modeling extinction, explosion, genetic bottlenecks, sweepstakes reproduction, and quasi-stationarity in population sizes under demographic or environmental stochasticity.
• Statistical Physics and Probability: Scaling limits, reaction-diffusion systems, and connections to extreme value theory; emergence of non-classical limit laws in heavy-tailed settings.
• Ecology and Epidemiology: Outbreaks, metapopulation dynamics; the dominance of "super-spreader" or "super-individuals" in transmission networks.
• Mathematical Finance: Cox-Ingersoll-Ross-like models with stochastic volatility and random environmental influences.

Immigration with heavy tails induces extremal (non-central) limiting processes, requiring sophisticated renormalizations distinct from classical diffusion limits (Foucart et al., 2023). The inclusion of collisions, state-dependent parameters, and non-Markovian time-changes continues to enlarge the modeling scope while preserving a robust stochastic-analytic foundation.

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Selected Mathematical Objects (for reference):

Object	Mathematical Formulation	Context
Branching mechanism $\Psi$	$$bq + \sigma^2 q^2 + \int_{(0,\infty)} [e^{-qz} - 1 + qz]\pi(dz)$$	Evolution generator
Immigration mechanism $\Phi$	$\beta q + \int_{(0,\infty)} (1 - e^{-qz}) \nu(dz)$	Immigration dynamics
Laplace transform	$$\mathbb{E}_x[e^{-\lambda X_t}] = \exp\{-x v_t(\lambda)\}$$	Law characterization
Lamperti transform	$Y_t := X_{T^{-1}(t)}, \, T(t) = \int_0^t Y_s ds$	Lévy process correspondence
Scale function $W$	$$\int_0^\infty e^{-\lambda x} W(x) dx = 1/\psi(\lambda)$$	Stationary measure, genealogy
Dirichlet form $\mathcal{E}$	$-\int (L f)(x) g(x) v(dx)$	Symmetry, sector condition

All presented statements and constructions are covered in detail in (Li, 2012, Handa, 2011, Murillo-Salas et al., 2015, Li, 2016, Palau et al., 2015, Pardo et al., 2013, Foucart et al., 2016, Andreis et al., 2017, Foucart et al., 2018, Fang et al., 2020, Liu et al., 2023, Abraham et al., 2020, Foucart et al., 2020, Foucart et al., 2022, Foucart et al., 2023, Li et al., 5 Feb 2025, Li et al., 6 Feb 2025). These works provide rigorous accounts of existence, uniqueness, limit theorems, path properties, and connections to martingale problems, scale functions, extremal processes, and genealogical trees in the full landscape of continuous-state branching models.