Controlled Branching Processes

• Controlled branching processes are discrete-time models where reproduction is regulated by a population-size–dependent control function, incorporating immigration, emigration, and resource constraints.
• They extend classical Galton–Watson models by introducing a controlled selection stage that alters criticality conditions, thereby influencing extinction probabilities and growth regimes.
• Recent advances include diffusion approximations, scaling limits, and sophisticated statistical inference methods that bridge discrete models with continuous analogues in branching systems.

Controlled branching processes are branching models in which reproduction is regulated by an explicit control mechanism rather than being purely autonomous at the individual level. In the standard one-type formulation, the population satisfies

$$Z_{n+1}=\sum_{j=1}^{\phi_n(Z_n)} X_{n,j},\qquad n\ge 0,$$

where the offspring variables $\{X_{n,j}\}$ are i.i.d., while $\phi_n(Z_n)$ is a random, population-size–dependent number of progenitors selected to reproduce in generation $n$ (González et al., 2021). In the broader terminology of the survey literature, controlled branching processes (CBPs), also called $\varphi$-branching processes, are discrete-time models in which the number of individuals allowed to reproduce in generation $n$ is itself a random, population-size–dependent quantity (Yanev, 2014). This framework includes Galton–Watson processes as the trivial case $\phi_n(k)=k$, but it also accommodates immigration, emigration, migration, resource limitation, state-dependent intervention, and continuous-state or spatial generalizations (Yanev, 2014).

1. Formal framework and principal model classes

A general one-type CBP starts from $Z_0\ge 0$ and uses two independent sources of randomness: an offspring law $\xi$ and a control family $\phi_n(k)$. Conditioned on $\{X_{n,j}\}$0, the next generation is obtained by first drawing the number of progenitors $\{X_{n,j}\}$1, and then summing $\{X_{n,j}\}$2 i.i.d. offspring counts (González et al., 2021). In the more general formulation used in the survey, one may index several control components by a set $\{X_{n,j}\}$3 and write

$\{X_{n,j}\}$4

so that the control can simultaneously represent existing individuals, immigrants, or other regulated inputs (Yanev, 2014).

The central structural feature is that population-size dependence enters through the number of active parents, not through a state-dependent modification of the offspring law itself (Yanev, 2014). This distinction separates CBPs from population-size–dependent branching processes, although the two classes can coincide under suitable equivalence conditions (Braunsteins et al., 2023). A controlled process may therefore be read as a Galton–Watson reproduction scheme preceded by a selection, augmentation, or truncation stage.

Several canonical controls recur across the literature. A process with immigration arises by taking $\{X_{n,j}\}$5, where $\{X_{n,j}\}$6 are i.i.d. immigration variables, so immigrants reproduce in the same generation as they arrive and with the same offspring law as resident individuals (González et al., 2021). State-dependent immigration appears in models with

$\{X_{n,j}\}$7

so immigration occurs only when the population is small (Yanev, 2014). Migration-type controls also appear, for example through a random variable $\{X_{n,j}\}$8 that reduces or increases the number of active parents and adds migrants when $\{X_{n,j}\}$9 (Yanev, 2014).

A particularly influential additive control is

$\phi_n(Z_n)$0

where $\phi_n(Z_n)$1 is a non-negative integer-valued random variable depending on the current population size $\phi_n(Z_n)$2 (Liu, 2022). In this construction, the additional term $\phi_n(Z_n)$3 behaves as immigration-like reproduction, but because it depends on the state, it generates dependent rather than independent immigration in the scaling limit (Liu, 2022).

2. Criticality, extinction, and growth regimes

For one-type CBPs, a basic asymptotic descriptor is the mean growth rate per individual in a population with $\phi_n(Z_n)$4 mothers,

$\phi_n(Z_n)$5

In the Galton–Watson case $\phi_n(Z_n)$6, $\phi_n(Z_n)$7 is constant, but for CBPs the sequence $\phi_n(Z_n)$8 can depend on $\phi_n(Z_n)$9, and its large-$n$0 behavior governs extinction and growth (Yanev, 2014).

The standard classification is asymptotic. A CBP is subcritical if $n$1, supercritical if $n$2, and critical if

$n$3

(Yanev, 2014). Under mild assumptions, if $n$4, then extinction occurs almost surely from every initial state, whereas if $n$5, there exists $n$6 such that extinction has probability strictly less than one for all initial populations $n$7 (Yanev, 2014). Two features distinguish CBPs from ordinary Galton–Watson theory: in supercritical CBPs, extinction may still occur almost surely when the initial population is too small, and in critical CBPs extinction probability need not equal one (Yanev, 2014).

The survey develops a finer critical theory through the stochastic difference equation

$n$8

with $n$9, and shows that in the asymptotically critical regime $\varphi$0, the balance between drift and conditional variance determines whether extinction is almost sure or whether escape to infinity occurs with positive probability (Yanev, 2014). This suggests that “criticality” in CBPs is not exhausted by first moments; higher-order fluctuation structure can be decisive.

Migration processes provide an explicit instance of this phenomenon. In the critical migration model with reflection at zero,

$\varphi$1

controls the long-run behavior, where $\varphi$2 is mean immigration outside zero, $\varphi$3 is emigration intensity, and $\varphi$4 (Yanev, 2014). Under irreducibility and aperiodicity, the process is non-recurrent for $\varphi$5, null-recurrent for $\varphi$6, and positive-recurrent for $\varphi$7 (Yanev, 2014). Conditional limit laws range from gamma limits to exponential Yaglom-type limits, depending on whether immigration dominates strongly or not (Yanev, 2014).

The inhomogeneous controlled branching process in varying environment (CPVE) adds a second layer of nonstationarity by letting the offspring distribution depend on generation $\varphi$8. In that model,

$\varphi$9

with generation-dependent offspring pgf $n$0 and control pgf $n$1, the process is a time-inhomogeneous Markov chain (González et al., 2024). Under $n$2 and $n$3, the classical duality persists:

$n$4

(González et al., 2024). Sufficient conditions for almost sure extinction and for positive survival probability are formulated in terms of the effective per-capita growth $n$5, where $n$6 and $n$7 (González et al., 2024).

3. Diffusion approximations and scaling limits

A major branch of the theory studies macroscopic limits obtained by rescaling space and time. For critical CBPs with random initial population, one works with

$n$8

and under the assumptions

$n$9

the scaled process converges in $\phi_n(k)=k$0 to a Feller diffusion with immigration,

$\phi_n(k)=k$1

(González et al., 2021). Here $\phi_n(k)=k$2 comes from the $\phi_n(k)=k$3 correction in the effective reproduction rate, while $\phi_n(k)=k$4 is the asymptotic diffusion coefficient inherited from offspring variance (González et al., 2021). The proof is based on martingale differences and limit theorems for random step processes rather than semigroup convergence (González et al., 2021).

A more general scaling limit, central to recent work, starts from an array of CBPs $\phi_n(k)=k$5 and the rescaled process

$\phi_n(k)=k$6

Under conditions (A–D) stated in terms of the offspring generating functions $\phi_n(k)=k$7 and control generating functions $\phi_n(k)=k$8, the process converges weakly in $\phi_n(k)=k$9 to a continuous-state branching process with dependent immigration (CBDI) (Liu, 2022). The limiting generator is characterized on Laplace test functions by

$Z_0\ge 0$0

where $Z_0\ge 0$1 has Lévy–Khintchine form and

$Z_0\ge 0$2

depends explicitly on the current state $Z_0\ge 0$3 (Liu, 2022). In the corresponding stochastic equation, the immigration drift is $Z_0\ge 0$4 and the immigration jump intensity is $Z_0\ge 0$5, so immigration is state dependent rather than independent (Liu, 2022).

This dependent-immigration picture is extended further when the control variable splits into an immigration term and a size-divisible term. In that setting,

$Z_0\ge 0$6

where $Z_0\ge 0$7 is immigration and $Z_0\ge 0$8 is $Z_0\ge 0$9-divisible, meaning it can be written as a sum of $\xi$0 i.i.d. random variables (González et al., 23 Aug 2025). Under sufficient conditions formulated through generating functions, the rescaled processes converge weakly on Skorokhod space to a continuous-state branching process with dependent immigration, and the size-divisible term contributes additional drift and diffusion components to the limit (González et al., 23 Aug 2025).

Multi-type diffusion approximation introduces a different geometry. For critical controlled multi-type branching processes with control expectations satisfying

$\xi$1

the scaled random step functions converge to a squared Bessel-type diffusion supported on the Perron–Frobenius ray of $\xi$2 (Barczy et al., 2023). Specifically,

$\xi$3

where $\xi$4 is the right Perron eigenvector and $\xi$5 solves a one-dimensional squared Bessel-type SDE (Barczy et al., 2023). Asymptotically, type frequencies converge to the coordinates of $\xi$6, while only the scalar mass fluctuates (Barczy et al., 2023).

The survey literature also records continuous-state CBPs defined directly in discrete time by

$\xi$7

where $\xi$8 is a counting process with stationary and independent increments, $\xi$9 are i.i.d. non-negative contributions, and $\phi_n(k)$0 is external input (Yanev, 2014). This formulation connects CBPs with continuous-state branching processes and with Galton–Watson processes with time-dependent immigration through an explicit duality (Yanev, 2014).

4. Spatial, measure-valued, and controlled-diffusion extensions

Once spatial motion is incorporated, CBPs enter the theory of branching diffusions and measure-valued control. In finite-horizon controlled branching diffusion, each particle moves according to a controlled diffusion

$\phi_n(k)$1

dies at rate $\phi_n(k)$2, and produces $\phi_n(k)$3 children with probability $\phi_n(k)$4 (Claisse, 2015). With a multiplicative cost functional,

$\phi_n(k)$5

the value function satisfies a branching property,

$\phi_n(k)$6

and the single-particle value function is the unique viscosity solution of a nonlinear Hamilton–Jacobi–Bellman equation (Claisse, 2015). This is a genuine optimal-control theory for branching systems in which motion and reproduction are both control-dependent (Claisse, 2015).

The interacting version replaces independent-particle coefficients by coefficients depending on the full current configuration. In that setting the state is a finite atomic measure on label–position space, admissible controls are predictable families indexed by particle labels, and dynamic programming leads to an infinite system of coupled HJB equations indexed by admissible particle configurations (Ocello, 16 Jan 2026). Under coercivity assumptions on the costs, growth bounds pass to the value function, a viscosity characterization is obtained, and a comparison principle gives uniqueness in the corresponding function class (Ocello, 16 Jan 2026). In the mean-field regime, permutation invariance of coefficients allows restriction to symmetric admissible controls (Ocello, 16 Jan 2026).

Another controlled-diffusion mechanism appears in the near-critical catalyst–reactant model with controlled immigration. There the catalyst is replenished to a threshold whenever it would fall below that threshold, while the reactant branches at a rate proportional to the current catalyst mass (Budhiraja et al., 2012). After scaling time and population size by $\phi_n(k)$7, the catalyst converges to a reflected diffusion on $\phi_n(k)$8, and the reactant converges to a diffusion whose coefficients depend on both catalyst and reactant (Budhiraja et al., 2012). In a fast-catalyst regime, stochastic averaging yields a one-dimensional effective SDE for the reactant, with coefficients determined by the invariant distribution of the reflected catalyst diffusion (Budhiraja et al., 2012). This suggests that threshold-based control in discrete branching models can appear macroscopically as reflection at a boundary.

At a still higher level of generality, branching Markov processes on configuration spaces can be controlled by a branching kernel $\phi_n(k)$9 and a killing function $\{X_{n,j}\}$00. On the finite-configuration space

$\{X_{n,j}\}$01

the process evolves by moving each particle according to a base Markov process, killing it at rate $\{X_{n,j}\}$02, and replacing it by a random finite configuration distributed by $\{X_{n,j}\}$03 (Beznea et al., 2015). This yields a branching standard process on $\{X_{n,j}\}$04, and when the base process is itself a superprocess, one obtains a discrete branching process on finite configurations of finite measures (Beznea et al., 2015). The control is encoded analytically by the nonlinear operator

$\{X_{n,j}\}$05

and probabilistically by the branching and killing kernels (Beznea et al., 2015).

5. Statistical inference, model choice, and estimability

Inference for CBPs is shaped by the observation scheme. When the full family tree is observed, including counts $\{X_{n,j}\}$06 of progenitors in generation $\{X_{n,j}\}$07 with exactly $\{X_{n,j}\}$08 offspring, the empirical offspring distribution

$\{X_{n,j}\}$09

is available, where $\{X_{n,j}\}$10 is the total number of progenitors producing $\{X_{n,j}\}$11 offspring up to generation $\{X_{n,j}\}$12, and $\{X_{n,j}\}$13 is the total number of progenitors up to generation $\{X_{n,j}\}$14 (González et al., 2018). On this basis, disparity-based posterior densities are defined by replacing the log-likelihood with $\{X_{n,j}\}$15, leading to expectation a D-posteriori (EDAP) and maximum a D-posteriori (MDAP) estimators (González et al., 2018). Under regularity conditions, these estimators are consistent and efficient under the postulated model, and they remain robust to misspecification and aberrant outliers (González et al., 2018).

A different regime arises when only generation sizes and at least the number of progenitors of the last generation are observed. In that setting, the posterior distribution of the parameters is approximated without explicit likelihood calculations by a two-stage ABC procedure: first, an ABC algorithm for model choice estimates the unknown maximum progeny per individual, and second, an ABC rejection algorithm with a summary statistic and a post-processing adjustment approximates the posterior of the main CBP parameters (González et al., 2021). The summary statistic

$\{X_{n,j}\}$16

is designed so that its coordinates converge to quantities involving the threshold parameter $\{X_{n,j}\}$17, the asymptotic control coefficient $\{X_{n,j}\}$18, and the offspring mean $\{X_{n,j}\}$19 (González et al., 2021).

A third line of work asks what can be consistently estimated from a single supercritical trajectory. For the supercritical CBP

$\{X_{n,j}\}$20

the answer depends sharply on what is observed (Braunsteins et al., 4 Apr 2025). If the control distribution is known, the offspring mean $\{X_{n,j}\}$21 and variance $\{X_{n,j}\}$22 are consistently estimable from a single trajectory of population counts, but finer offspring-law features are not (Braunsteins et al., 4 Apr 2025). If the control is unknown and only population sizes are observed, then under linear-moment control $\{X_{n,j}\}$23, $\{X_{n,j}\}$24, the estimable quantities are only

$\{X_{n,j}\}$25

not the full parameter vector $\{X_{n,j}\}$26 (Braunsteins et al., 4 Apr 2025). If progenitor numbers are observed alongside population sizes, then $\{X_{n,j}\}$27 become consistently estimable (Braunsteins et al., 4 Apr 2025). A common misconception is therefore that long supercritical trajectories always identify the underlying demographic and control parameters; the 2025 estimability results show that this is false without additional observation structure (Braunsteins et al., 4 Apr 2025).

The relation between CBPs and population-size–dependent branching processes (PSDBPs) further complicates model identification. The 2023 comparison paper proves conditions for exact equivalence, especially for deterministically controlled branching processes, and establishes an upper bound on the total variation distance between non-equivalent DCBPs and PSDBPs with matching first and second moments and equal initial population size (Braunsteins et al., 2023). Under certain conditions, this bound tends to zero as the initial population size becomes large (Braunsteins et al., 2023). This suggests that in large-population logistic regimes, model choice between CBP and PSDBP may be statistically delicate even when the two models are not exactly equivalent.

6. Conceptual issues, misconceptions, and research directions

Several points in the literature guard against oversimplified readings of the subject. First, a CBP is not merely a branching process with a modified offspring law. Its distinctive mechanism is control over the number of progenitors, whether deterministic, random, state dependent, or environment driven (Yanev, 2014). Second, the absorbing role of zero is model dependent. In many classical CBPs one assumes $\{X_{n,j}\}$28, but CBPs allowing immigration at zero are explicitly discussed, and such processes cannot be equivalent to PSDBPs with absorbing state $\{X_{n,j}\}$29 (Braunsteins et al., 2023). Third, “critical” does not imply almost sure extinction in the controlled setting, and “supercritical” does not preclude extinction from small initial states (Yanev, 2014).

A unifying theme of modern theory is the correspondence between discrete generating functions and continuous branching mechanisms. In the CBDI scaling limit, discrete offspring and control generating functions $\{X_{n,j}\}$30 and $\{X_{n,j}\}$31 converge to mechanisms $\{X_{n,j}\}$32 and $\{X_{n,j}\}$33 that enter both the generator and the stochastic equation of the limit (Liu, 2022). This discrete-to-continuous correspondence has now been extended to controls with size-divisible terms, to multitype processes with Perron–Frobenius geometry, and to reflected catalyst systems (González et al., 23 Aug 2025).

The survey literature explicitly records several open or only partially developed directions: branching processes with barriers, CBPs in random environments, alternating branching processes, a more detailed classification of critical CBPs for general $\{X_{n,j}\}$34 and variance structures, multitype controlled branching across types, spatial CBPs with migration and control, and a more comprehensive theory for continuous-state CBPs, including generators, scaling limits, and connections to CSBPs and measure-valued processes (Yanev, 2014). Recent work on viscosity solutions for interacting controlled branching diffusions and on finite-horizon optimal control shows that these directions are already interacting with mean-field control, stochastic target methods, and nonlinear PDE theory (Claisse, 2015, Ocello, 16 Jan 2026).

Taken together, the current literature presents controlled branching processes as a broad and technically heterogeneous family. At one end stand discrete one-type models with random control functions; at the other stand spatial, interacting, and measure-valued systems governed by coupled HJB equations or reflected diffusions. Across these settings, the defining question remains the same: how does regulation of reproductive participation alter extinction, growth, scaling, and inferential structure relative to the unconstrained Galton–Watson paradigm? The accumulated results indicate that this regulatory layer is not a minor perturbation but a source of genuinely new threshold phenomena, limit processes, and identifiability constraints (Yanev, 2014).