Interacting Branching Diffusion Processes

• Interacting branching diffusion processes are measure-valued models that combine stochastic branching, diffusive motion, and nonlinear pairwise interactions to simulate complex phenomena like extinction and phase transitions.
• They employ stochastic differential equations, mean-field limits, and discrete-to-continuum techniques to capture dynamics such as immigration, competition, and cooperation within diverse populations.
• The models yield rich dynamical behaviors, including traveling waves, Gaussian scaling limits, and genealogical structures, offering insights for control strategies and phase diagram analysis.

Interacting branching diffusion processes are measure-valued or particle system models in which individual agents (particles, individuals, types) exhibit both branching and diffusive spatial motion, together with nonlinear interactions driven by pairwise or type-level dependencies. The interplay between stochastic branching, spatial motion, immigration, and explicit competition or cooperation mechanisms produces a vast array of dynamical phenomena, including extinction, coexistence, phase transitions, nontrivial collective limits, and rich genealogical structures. These processes unify aspects of superprocesses, nonlinear interacting particle systems, mean-field diffusions, and structured Markov population models.

1. Fundamental Models and Stochastic Equations

The archetypal interacting branching diffusion—"branching diffusion with interactions"—describes populations evolving in $\mathbb{R}^d$ via dyadic branching and pairwise linear attraction or repulsion, possibly in the presence of Ornstein–Uhlenbeck (O–U) drift (Englander et al., 2016). For $2^m$ particles in $[m, m+1)$, each $Z^i_t$ solves: $$dZ^i_t = dW^i_t - b Z^i_t dt + \gamma \frac{1}{2^m}\sum_{j=1}^{2^m}(Z^j_t - Z^i_t)dt$$ with $W^i$ i.i.d. Brownian motions, drift $b$ (inward for $b>0$, outward for $b<0$), and interaction strength $\gamma\in\mathbb{R}$ (attraction if $\gamma>0$, repulsion if $\gamma<0$). The entire population is encoded as the atomic measure $Z_t = \sum_i \delta_{Z^i_t}$. Regular dyadic branching occurs at integer times.

Mean-field and multitype generalizations have been cast as multi-dimensional stochastic differential equations (SDEs) (Jin et al., 24 Dec 2025), incorporating diffusion, deterministic and stochastic immigration, linear and quadratic interaction matrices, and jump mechanisms. A CIMBI (Continuous-State Interacting Multi-Type Branching with Immigration) system is governed by,

$$X_i(t) = x_i + \int_0^t [\eta_i + \sum_{j=1}^d b_{ij} X_j(s) + \gamma_i(X(s)) ]ds + \int_0^t \sqrt{2\sigma_i X_i(s)} dW^i_s + \text{jump terms},$$

with quadratic interactions $\gamma_i(x) = \sum_j c_{ij} x_i x_j$, diffusion coefficients $\sigma_i$, and jump terms for immigration and branching.

Discrete-state counterparts and generalized Lamperti representations underpin connections between Markov chains (DIMBP: Discrete Interacting Multi-Type Branching Process) and their scaling limits, yielding interacting measure-valued diffusions under fine space/time normalization (Fittipaldi et al., 2022).

2. Macroscopic Behavior: Scaling Limits and Phase Diagrams

Long-term and large-population behavior critically depends on drift-interaction parameters, branching rate, and noise levels, manifesting in survival/extinction and scaling limit dichotomies. For the O–U branching-interaction system (Englander et al., 2016):

• Center of Mass (C.O.M.): For inward drift ($b>0$), the C.O.M. $\bar Z_t$ converges to the origin almost surely for any $\gamma$; for outward drift ($b<0$), $e^{bt}\bar Z_t$ converges to a Gaussian limit.
• SLLN: For noninteracting inward drift ($\gamma=0$, $b>0$),

$$2^{-n} Z_n(B) \longrightarrow \Bigl(\tfrac{b}{\pi}\Bigr)^{d/2} \int_B e^{-b|y|^2}dy, \quad \text{a.s.}$$

Gaussian asymptotics arise under appropriate mass renormalization.

• Phase diagram: System behavior splits into regions by $(b,\gamma)$: | Region | Parameter Regime | Outcome | |--------------|----------------------------------|-------------------------------------| | I | $b>0, b+\gamma>0$ | Gaussian scaling limit | | II | $b>0, b+\gamma=0$ | Lebesgue scaling after renormalization | | III | $b>0, b+\gamma<0$ | Local extinction vs. mass-growth dichotomy | | IV | $b<0, \forall \gamma$ | Outward drift: extinction |

In multi-type CIMBI models (Jin et al., 24 Dec 2025), sufficient immigration counteracts diffusive extinction; if $\eta_i>\sigma_i$ for all $i$, persistence occurs almost surely, while insufficient immigration ($\eta_i\leq \sigma_i/2$) and “sufficiently negative” quadratic interactions induce boundary hitting/extinction with positive probability.

3. Interaction Structures: Cooperation, Competition, and Nonlinearity

Interactions may be cooperative ($c_{ij}>0$) or competitive ($c_{ij}<0$), governed by linear and nonlinear terms:

• Linear interactions: $b_{ij}$ controls rates at which type $j$ boosts or suppresses type $i$.
• Quadratic interactions: $c_{ij} x_i x_j$ reflects intensity proportional to pairwise population masses, distinguishing between pure competition, pure cooperation, and mixed regimes (Jin et al., 24 Dec 2025).

Discrete interacting multitype models (DIMBP) (Fittipaldi et al., 2022) combine component-wise reproduction (random walks with Lamperti clock), and pairwise interactions driven by independent Poisson clocks with time-change $\int_0^t |c_{ij}| Z^i_s Z^j_s ds$. The generator is quartic, encoding nonlinear effects.

Mean-field limits further recast the dynamics as McKean–Vlasov type branching diffusions, where interaction terms depend on the empirical measure (Claisse et al., 2024, Claisse et al., 29 Nov 2025), yielding nonlinear Fokker–Planck or master equations for the measure-valued state.

4. Quantitative Convergence, Propagation of Chaos, and Limiting Equations

Recent progress includes explicit convergence rates for empirical measures toward nonlinear mean-field PDE limits (Fontbona et al., 2021), propagating chaos results, and stability in high-dimensional, measure-valued settings.

For logistic branching diffusions (Fontbona et al., 2021), under suitable initial and kinetic regularity assumptions: $$\sup_{t \in [0,T]} E \| \mu^K_t - \mu_t \|_{BL^*} \leq C_T [ I_4(K) + R_{d,q}(K) ],$$ where $I_4$ quantifies initial mass deviation and $R_{d,q}(K)$ is the Wasserstein convergence rate for $K$ samples.

The general McKean–Vlasov branching diffusion (Claisse et al., 2024) admits existence and uniqueness via contraction mapping, has well-posed weak and strong solution frameworks, and martingale problems that characterize propagation of chaos as $N \rightarrow \infty$.

Optimal control in interacting branching diffusions is formulated via dynamic programming, yielding infinite-dimensional HJB master equations on the space of finite nonnegative measures (Claisse et al., 29 Nov 2025, Ocello, 16 Jan 2026).

5. Boundary Behavior, Extinction, and Persistence Criteria

Boundary hitting (extinction or persistence) is critically determined by the balance of diffusion and immigration, as well as the sign and magnitude of interaction coefficients (Jin et al., 24 Dec 2025):

• Persistence: If $\eta_i > \sigma_i$ for all $i$, the process remains strictly inside $\mathbb{R}_+^d$.
• Extinction: If $\eta_i \leq \sigma_i/2$ and competition is strong ($c_{ij}\leq 0$), boundary is hit almost surely.

Foster–Lyapunov techniques and generator analysis are central to these results. For processes with jumps of finite activity, boundary is reached with positive probability under similar parameter regimes.

6. Genealogical and Population-Structured Representations

The genealogy of interacting branching diffusions admits contour-process and Ray–Knight representations, particularly in discrete-to-continuum limits (Ba et al., 2013):

• For nonlinear interaction $f$, population size evolves as:

$$Z^x_t = x + \int_0^t f(Z^x_s) ds + 2\int_0^t \int_0^{Z^x_s} W(ds,du)$$

• The generalized Ray–Knight theorem links the dynamics to the local times of a self-interacting reflected Brownian motion, representing population as local-time accumulations.

Competition/cooperation is interpreted directly through the gradient of local-time drift; extinction in the continuous mass limit is characterized by scale function analysis, generalizing Feller diffusion extinction criteria.

7. Special Classes: Mutually Catalytic Branching, Rank Interactions, and Front Propagation

Mutually catalytic branching processes, where branching rates are proportional to the product of the species masses, feature phase transitions dictated by random-walk recurrence versus transience and negative noise correlations. For example, with negative correlation (Doering et al., 2011), coexistence (versus extinction) is equivalent to transience of the underlying walk.

Infinite-rate counterparts on the boundary of the quadrant arise as scaling limits under time-rescaling and finite-system schemes, converging jointly in both coordinate and macroscopic averages (Doering et al., 2015).

Rank-based branching diffusions ("Go or Grow") with position-dependent branching and drift (Demircigil et al., 13 May 2025) exhibit explicit traveling wave phenomena. The minimal front speed transitions from “pulled” ($c^*=2$) to “pushed” ($c^* = \chi + 1/\chi$ for $\chi>1$) based on drift intensity, and ancestral lineage statistics reflect the wave type.

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References

• (Englander et al., 2016) Engländer & Zhang – Branching diffusion with interactions
• (Jin et al., 24 Dec 2025) Boundary behavior of CIMBI processes
• (Fittipaldi et al., 2022) On multitype Branching Processes with Interaction
• (Fontbona et al., 2021) Quantitative mean-field limit for interacting branching diffusions
• (Claisse et al., 2024) On McKean-Vlasov Branching Diffusion Processes
• (Claisse et al., 29 Nov 2025) Optimal Control of McKean--Vlasov Branching Diffusion Processes
• (Ocello, 16 Jan 2026) Controlled Interacting Branching Diffusion Processes: A Viscosity Approach
• (Ba et al., 2013) Branching processes with competition and generalized Ray Knight Theorem
• (Demircigil et al., 13 May 2025) Branching Rank-Based Interacting Brownian Particles
• (Doering et al., 2011) Longtime Behavior for Mutually Catalytic Branching with Negative Correlations
• (Doering et al., 2015) Finite system scheme for infinite rate mutually catalytic branching