Interacting Branching Diffusions

• Interacting branching diffusion processes are stochastic models that couple spatial motion, demographic branching, and explicit interactions at both microscopic and macroscopic levels.
• They employ diverse frameworks such as rank-based, McKean–Vlasov, and continuous-state multitype models to derive hydrodynamic limits linking particle dynamics to nonlinear PDEs.
• Analytical tools ranging from martingale problems to dynamic programming provide insights into front propagation, coexistence, and long-term behavior in these systems.

Interacting branching diffusion processes are stochastic population models in which spatial motion, demographic branching, and interaction are coupled at either the particle level or the level of empirical or marginal measures. In the recent arXiv literature, the term covers several technically distinct but structurally related classes: branching Brownian particles with rank-dependent motion and reproduction, McKean–Vlasov branching diffusions whose coefficients depend on the law of the alive population, continuous-state multitype branching diffusions with quadratic inter-specific interaction, mutually catalytic systems driven by correlated noises, and controlled measure-valued branching systems formulated through dynamic programming and Hamilton–Jacobi–Bellman equations (Demircigil et al., 13 May 2025, Claisse et al., 2024, Fittipaldi et al., 2022, Doering et al., 2011, Claisse et al., 29 Nov 2025, Ocello, 16 Jan 2026).

1. Model classes and state descriptions

A first major class is the microscopic particle system. In the rank-based branching Brownian model of 2025, one fixes an integer $K \ge 1$ and follows a population of particles $\{X^i_t\}_{i\in V_t}$ on $\mathbb R$, ranked so that the rightmost particle has rank $1$, the next rank $2$, and so on. Between branching times each particle evolves by Brownian motion, but particles of rank $>K$ receive an additional positive drift $\chi>0$, while only the $K$ rightmost particles branch, each at total rate $1$ (Demircigil et al., 13 May 2025). A related but distinct rank-based family keeps the population size fixed at $N$: particles move as independent Brownian motions, births occur at total rate $\{X^i_t\}_{i\in V_t}$0, and at each event one particle is duplicated while another is killed according to a rank-dependent selection density $\{X^i_t\}_{i\in V_t}$1 on $\{X^i_t\}_{i\in V_t}$2 (Mercer, 6 May 2026).

A second class is measure-dependent branching diffusion in the sense of McKean. In that formulation, particles are indexed by the Ulam–Harris tree and evolve according to SDEs whose drift, diffusion, branching rate, and offspring distribution depend on the marginal measure induced by all alive particles in the system. The empirical alive-particle measure is

$\{X^i_t\}_{i\in V_t}$3

while the interaction enters through the marginal

$\{X^i_t\}_{i\in V_t}$4

and each alive particle branches with rate $\{X^i_t\}_{i\in V_t}$5 and progeny law $\{X^i_t\}_{i\in V_t}$6 depending on $\{X^i_t\}_{i\in V_t}$7 in the controlled setting (Claisse et al., 29 Nov 2025). The uncontrolled McKean–Vlasov version allows coefficients to depend on the path and on the law of the branching system itself, and admits strong and weak formulations together with an equivalent martingale-problem characterization (Claisse et al., 2024).

A third class is continuous-state multitype interaction. In the multitype branching models motivated by stochastic Lotka–Volterra systems, the state is a vector $\{X^i_t\}_{i\in V_t}$8 of nonnegative masses, with drift terms of the form

$\{X^i_t\}_{i\in V_t}$9

diffusion coefficients $\mathbb R$0, and branching jumps generated by type-specific Lévy measures $\mathbb R$1 (Fittipaldi et al., 2022). Closely related continuous-state interacting multi-type branching processes with immigration include diffusion, immigration jumps, and type-dependent branching jumps, with interaction term

$\mathbb R$2

where the signs of $\mathbb R$3 encode competition, cooperation, or mixed interaction (Jin et al., 24 Dec 2025).

A fourth class consists of catalytic or symbiotic branching diffusions. In the negatively correlated two-type mutually catalytic system, one has site-indexed nonnegative populations $\mathbb R$4 and SDEs

$\mathbb R$5

$\mathbb R$6

with $\mathbb R$7 and $\mathbb R$8 (Doering et al., 2011). The finite-system scheme for infinite-rate mutually catalytic branching on $\mathbb R$9 sites instead uses a jump-type mean-field model on the axes of the first quadrant and identifies a rescaled diffusion limit for the total mass process (Doering et al., 2015).

These formulations differ in state space and interaction mechanism, but they share a common structural feature: branching is not autonomous. It is modulated by rank, by empirical occupation, by total or marginal mass, by pairwise interaction, or by spatial coexistence.

2. Microscopic dynamics and hydrodynamic limits

The microscopic rank-based model associated with the “Go or Grow” hypothesis couples movement and reproduction discontinuously. On each interval between branching times,

$1$0

so front particles “grow” without chemotactic drift, whereas particles behind the front “go” up the gradient with drift $1$1 but do not divide (Demircigil et al., 13 May 2025). The natural large-$1$2 object is the weighted empirical measure

$1$3

whose total mass is $1$4 and whose top-$1$5 particles carry mass exactly $1$6. For test functions $1$7, the martingale formulation identifies the drift operator

$1$8

and the martingale term has quadratic variation vanishing as $1$9 (Demircigil et al., 13 May 2025).

In the limit $2$0, any weak limit of $2$1 solves a nonlinear PDE for the density $2$2,

$2$3

or, in terms of the cumulative tail $2$4,

$2$5

The limiting equation is therefore free-boundary-like, with the threshold $2$6 separating the “Go” and “Grow” regimes (Demircigil et al., 13 May 2025).

The fixed-size $2$7-BBM yields a different hydrodynamic limit. If

$2$8

then under continuity and boundedness assumptions on $2$9 and $>K$0, the limit $>K$1 is the unique classical solution of

$>K$2

with nonlinearity $>K$3 (Mercer, 6 May 2026). The proof again proceeds through a martingale representation for smooth test functions, tightness in a Skorokhod space of measure-valued paths, identification of subsequential limits, and uniqueness of weak solutions.

These two hydrodynamic limits illustrate two distinct pathways from interacting particle systems to reaction–diffusion equations. In the rank-threshold “Go or Grow” system, the nonlinearity is generated by a moving mass threshold encoded by $>K$4. In the branching-selection system, the nonlinearity is generated by a selection density $>K$5 acting on ranks through the cumulative profile $>K$6 (Demircigil et al., 13 May 2025, Mercer, 6 May 2026).

3. Mean-field interaction and McKean–Vlasov branching

The McKean–Vlasov branching framework formalizes interaction through the law of the entire alive population rather than through explicit pairwise or rank-based coordinates. In the uncontrolled path-dependent formulation, each alive particle with label $>K$7 satisfies

$>K$8

dies at rate $>K$9, and is replaced by $\chi>0$0 offspring with probability $\chi>0$1 (Claisse et al., 2024). The self-consistency condition is that the environment measure equals the law of the branching process itself.

Under boundedness and Lipschitz assumptions in the path variable and in the Wasserstein-$\chi>0$2 distance, the fixed-point map $\chi>0$3 is a contraction on a small time interval, which yields strong existence and uniqueness. The same work also establishes a weak-solution theory and an equivalent martingale problem on the canonical space $\chi>0$4, and proves propagation of chaos: the empirical-law processes of an interacting $\chi>0$5-system are tight, and any limit point is supported on solutions of the McKean–Vlasov branching martingale problem (Claisse et al., 2024).

The controlled McKean–Vlasov model narrows the dependence to the current marginal measure $\chi>0$6 and to Lipschitz closed-loop controls $\chi>0$7. For a particle $\chi>0$8,

$\chi>0$9

and at branching times the expected net reproduction enters through

$K$0

The corresponding deterministic flow of marginal measures solves the nonlinear Fokker–Planck equation

$K$1

in the distributional sense (Claisse et al., 29 Nov 2025).

This measure-valued perspective is important because it separates the random genealogical system from a deterministic forward law. A plausible implication is that the nonlinear PDE is the natural object for both control and limit-theorem analysis: in the uncontrolled theory it identifies the mean-field limit, while in the controlled theory it is the state equation on the space of measures (Claisse et al., 2024, Claisse et al., 29 Nov 2025).

4. Multitype, competitive, cooperative, and catalytic interaction

Continuous-state multitype branching with interaction is designed to encode stochastic Lotka–Volterra-type mechanisms. In the formulation of Fittipaldi and Palau, the $K$2th component satisfies

$K$3

The model has a generalized Lamperti-type representation

$K$4

where the time changes involve both linear occupation and pairwise interaction terms (Fittipaldi et al., 2022). The same paper proves that suitably renormalized discrete interacting multitype branching processes converge in Skorokhod topology to this continuous-state limit.

A related diffusion-with-interaction model is the continuous-state interacting multi-type branching process with immigration. There the $K$5th component includes immigration rates $K$6, linear branching drift $K$7, diffusion term $K$8, immigration jumps, type-dependent branching jumps, and quadratic interaction

$K$9

with $1$0 and off-diagonal signs determining competition, cooperation, or mixed regimes (Jin et al., 24 Dec 2025). This formulation is explicitly aimed at boundary behavior in $1$1.

Discrete-state branching processes with interactions offer a parallel, non-diffusive perspective. The family introduced in 2017 allows individual births and deaths, pairwise competition, annihilation, cooperation, and catastrophe events. The key cooperative parameter is

$1$2

Under the subcritical cooperative regime $1$3 and $1$4, the process is non-explosive and comes down from infinity, and when $1$5 it admits a moment dual in $1$6 with

$1$7

(Casanova et al., 2017). This is a discrete analogue rather than a diffusion model, but it supplies a rigorous interaction taxonomy—competition, annihilation, cooperation, catastrophes—that reappears in continuous-state branching diffusions.

Catalytic interaction leads to a different mechanism. In the negatively correlated mutually catalytic model, branching noise is proportional to $1$8, so activity of one type catalyzes that of the other. For any finite or infinite branching rate, coexistence is possible if and only if the migration generator is transient: $1$9 Equivalently, coexistence is impossible if and only if the underlying migration chain is recurrent (Doering et al., 2011). In the infinite-rate mean-field setting, the finite-system scheme shows that the rescaled total mass process converges to the mutually catalytic branching diffusion

$N$0

after the logarithmic time rescaling $N$1 (Doering et al., 2015).

Together, these models show that “interaction” in branching diffusion theory is not a single mechanism. It may enter through quadratic drift, through catalytic noise, through state-dependent branching intensities, or through genealogically aggregated selection.

5. Long-time behavior, front propagation, and boundary phenomena

Long-time analysis is especially developed for rank-based branching Brownian fronts. For the hydrodynamic limit of the “Go or Grow” model, traveling-wave solutions $N$2 exhibit a minimal wave speed

$N$3

with a critical parameter $N$4 (Demircigil et al., 13 May 2025). In the pulled regime, the leading edge determines the speed and the front exhibits the Bramson logarithmic shifts

$N$5

whereas for $N$6 one has linear motion $N$7. Numerical study of ancestral lineages further distinguishes the two regimes: in the pulled case ancestry drifts toward the edge, while in the pushed case it settles to a steady profile (Demircigil et al., 13 May 2025).

The fixed-size branching-selection model produces an analogous connection between particle speeds and PDE wave speeds. Under the assumptions that $N$8, $N$9 on $\{X^i_t\}_{i\in V_t}$00, and $\{X^i_t\}_{i\in V_t}$01 on $\{X^i_t\}_{i\in V_t}$02, all extreme ranks travel with a single almost-sure asymptotic velocity

$\{X^i_t\}_{i\in V_t}$03

and as $\{X^i_t\}_{i\in V_t}$04,

$\{X^i_t\}_{i\in V_t}$05

Under the same assumptions, $\{X^i_t\}_{i\in V_t}$06, the limiting PDE has minimal traveling-wave speed $\{X^i_t\}_{i\in V_t}$07, and the particle speed converges to this minimal wave speed, yielding a partial weak-selection principle (Mercer, 6 May 2026).

Boundary and extinction behavior are central in multitype continuous-state models. For the CIMBI process, if $\{X^i_t\}_{i\in V_t}$08 and for each $\{X^i_t\}_{i\in V_t}$09 either $\{X^i_t\}_{i\in V_t}$10 or $\{X^i_t\}_{i\in V_t}$11 together with $\{X^i_t\}_{i\in V_t}$12, then

$\{X^i_t\}_{i\in V_t}$13

for each coordinate (Jin et al., 24 Dec 2025). By contrast, in the diffusion case with $\{X^i_t\}_{i\in V_t}$14 and $\{X^i_t\}_{i\in V_t}$15 for all $\{X^i_t\}_{i\in V_t}$16, together with a non-positive or negative-definite quadratic-form condition involving $\{X^i_t\}_{i\in V_t}$17, the process hits the boundary almost surely: $\{X^i_t\}_{i\in V_t}$18 With finite-activity jumps under analogous small-immigration conditions, boundary hitting has strictly positive probability (Jin et al., 24 Dec 2025).

Long-time behavior also appears in spatially interacting branching diffusions with linear attraction or repulsion and Ornstein–Uhlenbeck drift. In that model the center of mass satisfies

$\{X^i_t\}_{i\in V_t}$19

so $\{X^i_t\}_{i\in V_t}$20 almost surely when $\{X^i_t\}_{i\in V_t}$21, while for $\{X^i_t\}_{i\in V_t}$22 it escapes exponentially fast with rate $\{X^i_t\}_{i\in V_t}$23 (Englander et al., 2016). After centering, the relative system behaves like an Ornstein–Uhlenbeck motion with parameter $\{X^i_t\}_{i\in V_t}$24, which yields an SLLN in the inward regime $\{X^i_t\}_{i\in V_t}$25, critical polynomial scaling at $\{X^i_t\}_{i\in V_t}$26, and local extinction for bounded sets when $\{X^i_t\}_{i\in V_t}$27 (Englander et al., 2016).

These results show that interacting branching diffusion processes display several recurrent asymptotic dichotomies: pushed versus pulled fronts, coexistence versus extinction, persistence versus boundary hitting, and concentration versus local extinction.

6. Analytical methods and control formulations

The analytical toolkit is heterogeneous but highly structured. In the rank-based hydrodynamic limit, the main ingredients are the martingale problem for $\{X^i_t\}_{i\in V_t}$28, moment controls, tightness in Skorokhod space, discrete integration by parts on the empirical cumulative function, passage to the cumulative tail $\{X^i_t\}_{i\in V_t}$29, and a contraction argument on the moving threshold $\{X^i_t\}_{i\in V_t}$30 satisfying $\{X^i_t\}_{i\in V_t}$31 (Demircigil et al., 13 May 2025). In the branching-selection model, the proof of the hydrodynamic limit likewise uses a martingale representation for $\{X^i_t\}_{i\in V_t}$32, tightness of measure-valued paths, identification of the nonlinear term through the approximation $\{X^i_t\}_{i\in V_t}$33, and uniqueness by a duality or semigroup argument for the parabolic PDE (Mercer, 6 May 2026).

For McKean–Vlasov branching, fixed-point arguments and martingale problems are fundamental. Strong well-posedness follows from contraction of the frozen-environment map, while weak existence and propagation of chaos are obtained by tightness, Aldous-type criteria in Skorokhod space, and limit identification through martingale-problem convergence (Claisse et al., 2024). The controlled measure-valued theory then adds a dynamic programming structure. For running cost $\{X^i_t\}_{i\in V_t}$34 and terminal cost $\{X^i_t\}_{i\in V_t}$35, the value function

$\{X^i_t\}_{i\in V_t}$36

satisfies the dynamic programming principle

$\{X^i_t\}_{i\in V_t}$37

and, under regularity assumptions, the HJB master equation on $\{X^i_t\}_{i\in V_t}$38 (Claisse et al., 29 Nov 2025).

A complementary viscosity approach formulates controlled interacting branching diffusion processes as an infinite coupled HJB system on the disjoint union of Euclidean spaces corresponding to admissible particle configurations. If

$\{X^i_t\}_{i\in V_t}$39

then the HJB system is

$\{X^i_t\}_{i\in V_t}$40

with terminal condition $\{X^i_t\}_{i\in V_t}$41 (Ocello, 16 Jan 2026). Under coercivity conditions, growth bounds transfer from the cost functionals to the value function, which admits a viscosity characterization together with a comparison principle.

In the mean-field regime of that control problem, permutation invariance of the coefficients implies symmetry of the value function under relabeling, and measurable-selection arguments permit restriction to symmetric admissible controls (Ocello, 16 Jan 2026). This suggests a convergence of perspectives: particle-level HJB systems, measure-space master equations, and nonlinear Fokker–Planck equations are different representations of the same control-theoretic structure when interaction is mediated by the evolving population law.

A plausible implication is that interacting branching diffusion processes now sit at an interface between stochastic particle systems, nonlinear PDEs, and infinite-dimensional control. The cited works do not collapse these viewpoints into a single universal framework, but they do provide rigorous bridges: hydrodynamic limits from particles to PDEs, propagation of chaos from many-body systems to McKean–Vlasov laws, and dynamic programming principles from stochastic genealogies to HJB equations on spaces of configurations or measures (Demircigil et al., 13 May 2025, Claisse et al., 2024, Claisse et al., 29 Nov 2025, Ocello, 16 Jan 2026).