Branching Diffusion with Interactions

• Branching Diffusion with Interactions is a stochastic framework where individuals diffuse and branch based on interaction-dependent rules, capturing complex population dynamics.
• It incorporates various mechanisms such as rank-based interactions, mean-field effects, and catalytic branching to model both local and global behaviors.
• Analytical methods like hydrodynamic limits, traveling wave analysis, and spine techniques provide insights into long-term system behavior and control.

Searching arXiv for recent and foundational papers on interacting branching diffusions to ground the article. Branching diffusion with interactions denotes a class of stochastic population models in which individuals move in continuous space, typically by diffusion, while branching, death, or selection events depend on the current population configuration rather than on independent per-particle mechanisms alone. Recent formulations represent the population either by a finite atomic measure on an Ulam–Harris tree or by a weighted empirical measure, and place the interaction in rank, local occupancy, empirical-law dependence, pairwise catalytic terms, or nonlinear competition (Claisse et al., 2024, Ocello, 16 Jan 2026, Demircigil et al., 13 May 2025). In this sense, the subject includes rank-based branching Brownian systems, McKean–Vlasov branching diffusions, mutually catalytic branching, spatially self-regulating diffusions, and multitype interacting branching processes, together with their hydrodynamic, mean-field, and genealogical limits.

1. State spaces, labeling schemes, and martingale formulations

A common representation uses Ulam–Harris labels to encode genealogies. In the controlled interacting branching-diffusion framework, the label set is

$$\mathcal I_c=\{\varnothing\}\;\cup\;\bigcup_{n\ge1}\mathbb N^n,$$

and the state space is

$$E\;=\;\Bigl\{\lambda=\sum_{i\in\mathcal V}\delta_{(i,x_i)}:\mathcal V\subset\mathcal I_c\text{ finite},\;x_i\in\mathbb R^d\Bigr\},$$

endowed with the weak-* topology. A controlled branching-diffusion is then a càdlàg, adapted $E$-valued process satisfying a martingale problem in which each labeled particle evolves by a diffusion, branches according to a configuration-dependent death rate, and produces $k$ offspring with probability $p_k(i,x,\lambda,a)$ (Ocello, 16 Jan 2026).

An equivalent population-level notation appears in McKean–Vlasov branching diffusion. There one writes

$Z_t=\sum_{k\in\mathcal K_t}\delta_{(k,X^k_t)},$

with $\mathcal K_t$ the set of alive particles and $\mu_t=\mathrm{Law}(Z_t)$ the law of the whole branching process up to time $t$. Each particle follows

$$dX^k_t = b\bigl(t,X^k_{\cdot\wedge t},\mu_t\bigr)\,dt + \sigma\bigl(t,X^k_{\cdot\wedge t},\mu_t\bigr)\,dW^k_t,$$

dies at rate $$E\;=\;\Bigl\{\lambda=\sum_{i\in\mathcal V}\delta_{(i,x_i)}:\mathcal V\subset\mathcal I_c\text{ finite},\;x_i\in\mathbb R^d\Bigr\},$$0, and at death produces a random number of offspring with law $$E\;=\;\Bigl\{\lambda=\sum_{i\in\mathcal V}\delta_{(i,x_i)}:\mathcal V\subset\mathcal I_c\text{ finite},\;x_i\in\mathbb R^d\Bigr\},$$1 (Claisse et al., 2024).

For large but finite populations, weighted empirical measures are standard. In the rank-based Brownian system one records

$$E\;=\;\Bigl\{\lambda=\sum_{i\in\mathcal V}\delta_{(i,x_i)}:\mathcal V\subset\mathcal I_c\text{ finite},\;x_i\in\mathbb R^d\Bigr\},$$2

where $$E\;=\;\Bigl\{\lambda=\sum_{i\in\mathcal V}\delta_{(i,x_i)}:\mathcal V\subset\mathcal I_c\text{ finite},\;x_i\in\mathbb R^d\Bigr\},$$3 is the set of alive labels and $$E\;=\;\Bigl\{\lambda=\sum_{i\in\mathcal V}\delta_{(i,x_i)}:\mathcal V\subset\mathcal I_c\text{ finite},\;x_i\in\mathbb R^d\Bigr\},$$4 is the rescaled population size (Demircigil et al., 13 May 2025). In mean-field interacting branching diffusions with logistic binary branching one likewise writes

$$E\;=\;\Bigl\{\lambda=\sum_{i\in\mathcal V}\delta_{(i,x_i)}:\mathcal V\subset\mathcal I_c\text{ finite},\;x_i\in\mathbb R^d\Bigr\},$$5

so that total mass remains of order $$E\;=\;\Bigl\{\lambda=\sum_{i\in\mathcal V}\delta_{(i,x_i)}:\mathcal V\subset\mathcal I_c\text{ finite},\;x_i\in\mathbb R^d\Bigr\},$$6 under the $$E\;=\;\Bigl\{\lambda=\sum_{i\in\mathcal V}\delta_{(i,x_i)}:\mathcal V\subset\mathcal I_c\text{ finite},\;x_i\in\mathbb R^d\Bigr\},$$7-weighting (Fontbona et al., 2021).

Across these formulations, martingale problems provide the unifying analytic language. In the rank-based model, for each $$E\;=\;\Bigl\{\lambda=\sum_{i\in\mathcal V}\delta_{(i,x_i)}:\mathcal V\subset\mathcal I_c\text{ finite},\;x_i\in\mathbb R^d\Bigr\},$$8,

$$E\;=\;\Bigl\{\lambda=\sum_{i\in\mathcal V}\delta_{(i,x_i)}:\mathcal V\subset\mathcal I_c\text{ finite},\;x_i\in\mathbb R^d\Bigr\},$$9

with a local generator $E$0 and a martingale $E$1 whose quadratic variation is $E$2 (Demircigil et al., 13 May 2025). In the McKean–Vlasov setting, weak solutions are characterized by the martingale property of

$E$3

for smooth test functions $E$4, where $E$5 is a nonlinear generator acting on the current configuration (Claisse et al., 2024).

2. Principal interaction mechanisms

One major class is rank-based interaction. In the “Go or Grow” system, the rank of a particle in a configuration $E$6 is

$E$7

The $E$8 rightmost particles, with ranks $E$9, are in the Grow regime and branch at rate $k$0 each, while all others are in the Go regime, drift rightward with speed $k$1, and do not branch (Demircigil et al., 13 May 2025). The motion between branching events is Brownian with piecewise-constant drift,

$k$2

where $k$3 indicates whether the particle has rank $k$4 (Demircigil et al., 13 May 2025). The model is explicitly tied to the Go or Grow hypothesis for cells in a capillary tube moving upwards a chemical gradient.

A second class is mean-field interaction through smoothed empirical fields. In the quantitative mean-field model, each particle solves

$k$5

with bounded Lipschitz kernels $k$6, birth rate $k$7, and death rate $k$8 (Fontbona et al., 2021). The interaction therefore enters both the diffusion coefficients and the logistic mass regulation.

A third class is nonlinear dependence on the law of the whole branching process. In McKean–Vlasov branching diffusion, the coefficients $k$9, the death rate $p_k(i,x,\lambda,a)$0, and the offspring laws $p_k(i,x,\lambda,a)$1 depend on $p_k(i,x,\lambda,a)$2, so the interaction is measure-valued and path-dependent (Claisse et al., 2024). Under boundedness, measurability, and Lipschitz conditions in path and measure under $p_k(i,x,\lambda,a)$3, strong well-posedness is obtained (Claisse et al., 2024).

Catalytic interaction produces a different mechanism. In finite-rate two-type mutually catalytic branching on a countable site space $p_k(i,x,\lambda,a)$4,

$p_k(i,x,\lambda,a)$5

where $p_k(i,x,\lambda,a)$6 and $p_k(i,x,\lambda,a)$7 implements negative correlation (Doering et al., 2011). In the discrete particle counterpart on $p_k(i,x,\lambda,a)$8, a type-1 particle at $p_k(i,x,\lambda,a)$9 branches at instantaneous rate $Z_t=\sum_{k\in\mathcal K_t}\delta_{(k,X^k_t)},$0, and symmetrically a type-2 particle branches at rate $Z_t=\sum_{k\in\mathcal K_t}\delta_{(k,X^k_t)},$1, where the catalyst is the local population of the opposite type (Fugenfirov et al., 2023).

Competition and self-regulation give another canonical structure. In the small-mass diffusion limit of a multitype locally self-regulating model,

$Z_t=\sum_{k\in\mathcal K_t}\delta_{(k,X^k_t)},$2

so migration, local carrying capacities, and cross-type competition are all explicit in the drift (Greven et al., 2015). A related one-dimensional competition limit is

$Z_t=\sum_{k\in\mathcal K_t}\delta_{(k,X^k_t)},$3

where $Z_t=\sum_{k\in\mathcal K_t}\delta_{(k,X^k_t)},$4 encodes net births minus deaths including competition (Ba et al., 2013).

Finally, interaction may act through the center of mass. In the dyadic branching diffusion with Ornstein–Uhlenbeck drift and mean-field attraction or repulsion,

$Z_t=\sum_{k\in\mathcal K_t}\delta_{(k,X^k_t)},$5

so the overall drift is the combination of Ornstein–Uhlenbeck confinement or expansion and pairwise mean-field attraction or repulsion (Englander et al., 2016).

3. Large-population, hydrodynamic, and scaling limits

The rank-based Go or Grow system admits a hydrodynamic limit as $Z_t=\sum_{k\in\mathcal K_t}\delta_{(k,X^k_t)},$6. Writing $Z_t=\sum_{k\in\mathcal K_t}\delta_{(k,X^k_t)},$7, defining the cumulative tail

$Z_t=\sum_{k\in\mathcal K_t}\delta_{(k,X^k_t)},$8

and weighting individuals by $Z_t=\sum_{k\in\mathcal K_t}\delta_{(k,X^k_t)},$9, the limit is the nonlinear integro-PDE

$\mathcal K_t$0

or, in terms of $\mathcal K_t$1,

$\mathcal K_t$2

with tightness and convergence established under finite third-moment initial data and vague convergence of the initial empirical measures (Demircigil et al., 13 May 2025).

Mean-field interacting branching diffusions with logistic binary branching admit both qualitative and quantitative limits. Under i.i.d. initial atoms, Lipschitz coefficients, and bounded Lipschitz kernels, one has $\mathcal K_t$3 in $\mathcal K_t$4, where $\mathcal K_t$5 is the unique continuous finite-measure solution of

$\mathcal K_t$6

Its total mass $\mathcal K_t$7 solves the logistic ODE $\mathcal K_t$8 (Fontbona et al., 2021). With higher moments, the convergence rate is explicit: $\mathcal K_t$9 The proof uses a coupling with an auxiliary process of McKean–Vlasov diffusions and optimal transport at branching times (Fontbona et al., 2021).

McKean–Vlasov branching diffusion itself is obtained as a large-population limit. For the $\mu_t=\mathrm{Law}(Z_t)$0-particle interacting system in which coefficients depend on the empirical law, the law of the empirical measure is tight in $\mu_t=\mathrm{Law}(Z_t)$1, and any limit point is supported on solutions of the McKean–Vlasov martingale problem. If strong uniqueness holds, the entire sequence converges to the law of the unique McKean–Vlasov branching diffusion (Claisse et al., 2024).

Several interacting branching models admit discrete-to-continuous scaling limits. In the multitype interacting branching framework, if $\mu_t=\mathrm{Law}(Z_t)$2 is the discrete-state interacting multitype branching process and $\mu_t=\mathrm{Law}(Z_t)$3 satisfy the scaling assumptions, then

$\mu_t=\mathrm{Law}(Z_t)$4

where $\mu_t=\mathrm{Law}(Z_t)$5 is the unique strong solution of the continuous-state interacting branching SDE with quadratic interaction terms (Fittipaldi et al., 2022). In branching processes with competition, the renormalized population size converges to the nonlinear diffusion $\mu_t=\mathrm{Law}(Z_t)$6, and the whole family $\mu_t=\mathrm{Law}(Z_t)$7 is represented by local times of a reflected Brownian motion with local-time-dependent drift, yielding a generalized Ray–Knight theorem (Ba et al., 2013).

The finite-system scheme for infinite-rate mutually catalytic branching has a different time scale because the jumps lack finite second moments. For the mean-field site space $\mu_t=\mathrm{Law}(Z_t)$8, the correct scale for the total masses is

$\mu_t=\mathrm{Law}(Z_t)$9

and the rescaled averages $t$0 converge to

$t$1

with joint convergence of total masses and coordinate processes under the finite-system scheme (Doering et al., 2015).

4. Traveling waves, invasion speeds, and genealogical front classification

For the rank-based Go or Grow limit PDE, traveling waves are obtained from the ansatz $t$2, $t$3, with cumulative tail $t$4. The resulting ODE admits a minimal wave speed

$t$5

such that for each $t$6 there is a monotone profile $t$7 with $t$8 and $t$9, while no such wave exists for $$dX^k_t = b\bigl(t,X^k_{\cdot\wedge t},\mu_t\bigr)\,dt + \sigma\bigl(t,X^k_{\cdot\wedge t},\mu_t\bigr)\,dW^k_t,$$0 (Demircigil et al., 13 May 2025). The transition occurs at the critical value $$dX^k_t = b\bigl(t,X^k_{\cdot\wedge t},\mu_t\bigr)\,dt + \sigma\bigl(t,X^k_{\cdot\wedge t},\mu_t\bigr)\,dW^k_t,$$1. When $$dX^k_t = b\bigl(t,X^k_{\cdot\wedge t},\mu_t\bigr)\,dt + \sigma\bigl(t,X^k_{\cdot\wedge t},\mu_t\bigr)\,dW^k_t,$$2, the front is pulled and $$dX^k_t = b\bigl(t,X^k_{\cdot\wedge t},\mu_t\bigr)\,dt + \sigma\bigl(t,X^k_{\cdot\wedge t},\mu_t\bigr)\,dW^k_t,$$3 coincides with the linear spreading speed of $$dX^k_t = b\bigl(t,X^k_{\cdot\wedge t},\mu_t\bigr)\,dt + \sigma\bigl(t,X^k_{\cdot\wedge t},\mu_t\bigr)\,dW^k_t,$$4. When $$dX^k_t = b\bigl(t,X^k_{\cdot\wedge t},\mu_t\bigr)\,dt + \sigma\bigl(t,X^k_{\cdot\wedge t},\mu_t\bigr)\,dW^k_t,$$5, the front is pushed and $$dX^k_t = b\bigl(t,X^k_{\cdot\wedge t},\mu_t\bigr)\,dt + \sigma\bigl(t,X^k_{\cdot\wedge t},\mu_t\bigr)\,dW^k_t,$$6 (Demircigil et al., 13 May 2025).

Microscopic finite-$$dX^k_t = b\bigl(t,X^k_{\cdot\wedge t},\mu_t\bigr)\,dt + \sigma\bigl(t,X^k_{\cdot\wedge t},\mu_t\bigr)\,dW^k_t,$$7 simulations use the position of the $$dX^k_t = b\bigl(t,X^k_{\cdot\wedge t},\mu_t\bigr)\,dt + \sigma\bigl(t,X^k_{\cdot\wedge t},\mu_t\bigr)\,dW^k_t,$$8-th largest particle,

$$dX^k_t = b\bigl(t,X^k_{\cdot\wedge t},\mu_t\bigr)\,dt + \sigma\bigl(t,X^k_{\cdot\wedge t},\mu_t\bigr)\,dW^k_t,$$9

and the empirical speed estimator

$$E\;=\;\Bigl\{\lambda=\sum_{i\in\mathcal V}\delta_{(i,x_i)}:\mathcal V\subset\mathcal I_c\text{ finite},\;x_i\in\mathbb R^d\Bigr\},$$00

For each fixed $$E\;=\;\Bigl\{\lambda=\sum_{i\in\mathcal V}\delta_{(i,x_i)}:\mathcal V\subset\mathcal I_c\text{ finite},\;x_i\in\mathbb R^d\Bigr\},$$01, $$E\;=\;\Bigl\{\lambda=\sum_{i\in\mathcal V}\delta_{(i,x_i)}:\mathcal V\subset\mathcal I_c\text{ finite},\;x_i\in\mathbb R^d\Bigr\},$$02 grows roughly linearly in $$E\;=\;\Bigl\{\lambda=\sum_{i\in\mathcal V}\delta_{(i,x_i)}:\mathcal V\subset\mathcal I_c\text{ finite},\;x_i\in\mathbb R^d\Bigr\},$$03. As $$E\;=\;\Bigl\{\lambda=\sum_{i\in\mathcal V}\delta_{(i,x_i)}:\mathcal V\subset\mathcal I_c\text{ finite},\;x_i\in\mathbb R^d\Bigr\},$$04 grows, $$E\;=\;\Bigl\{\lambda=\sum_{i\in\mathcal V}\delta_{(i,x_i)}:\mathcal V\subset\mathcal I_c\text{ finite},\;x_i\in\mathbb R^d\Bigr\},$$05 concentrates around a deterministic limit, and numerically

$$E\;=\;\Bigl\{\lambda=\sum_{i\in\mathcal V}\delta_{(i,x_i)}:\mathcal V\subset\mathcal I_c\text{ finite},\;x_i\in\mathbb R^d\Bigr\},$$06

with small finite-$$E\;=\;\Bigl\{\lambda=\sum_{i\in\mathcal V}\delta_{(i,x_i)}:\mathcal V\subset\mathcal I_c\text{ finite},\;x_i\in\mathbb R^d\Bigr\},$$07 corrections in the pulled regime $$E\;=\;\Bigl\{\lambda=\sum_{i\in\mathcal V}\delta_{(i,x_i)}:\mathcal V\subset\mathcal I_c\text{ finite},\;x_i\in\mathbb R^d\Bigr\},$$08, reflecting the Bramson logarithmic shift (Demircigil et al., 13 May 2025).

The same model supports a genealogical criterion for pulled versus pushed fronts. If a particle sampled at time $$E\;=\;\Bigl\{\lambda=\sum_{i\in\mathcal V}\delta_{(i,x_i)}:\mathcal V\subset\mathcal I_c\text{ finite},\;x_i\in\mathbb R^d\Bigr\},$$09 is followed backward along its unique ancestral lineage, and the lineage is observed in the moving frame of the front, then Conjecture 5.5 states that for large $$E\;=\;\Bigl\{\lambda=\sum_{i\in\mathcal V}\delta_{(i,x_i)}:\mathcal V\subset\mathcal I_c\text{ finite},\;x_i\in\mathbb R^d\Bigr\},$$10

$$E\;=\;\Bigl\{\lambda=\sum_{i\in\mathcal V}\delta_{(i,x_i)}:\mathcal V\subset\mathcal I_c\text{ finite},\;x_i\in\mathbb R^d\Bigr\},$$11

In the pushed case $$E\;=\;\Bigl\{\lambda=\sum_{i\in\mathcal V}\delta_{(i,x_i)}:\mathcal V\subset\mathcal I_c\text{ finite},\;x_i\in\mathbb R^d\Bigr\},$$12, the adjoint operator admits a unique integrable invariant density

$$E\;=\;\Bigl\{\lambda=\sum_{i\in\mathcal V}\delta_{(i,x_i)}:\mathcal V\subset\mathcal I_c\text{ finite},\;x_i\in\mathbb R^d\Bigr\},$$13

and numerically the lineage histogram closely matches $$E\;=\;\Bigl\{\lambda=\sum_{i\in\mathcal V}\delta_{(i,x_i)}:\mathcal V\subset\mathcal I_c\text{ finite},\;x_i\in\mathbb R^d\Bigr\},$$14. In the pulled case $$E\;=\;\Bigl\{\lambda=\sum_{i\in\mathcal V}\delta_{(i,x_i)}:\mathcal V\subset\mathcal I_c\text{ finite},\;x_i\in\mathbb R^d\Bigr\},$$15, the formal stationary solution is not integrable, so the lineage law drifts to $$E\;=\;\Bigl\{\lambda=\sum_{i\in\mathcal V}\delta_{(i,x_i)}:\mathcal V\subset\mathcal I_c\text{ finite},\;x_i\in\mathbb R^d\Bigr\},$$16; ancestors come only from the ever-advancing tip. Exactly at $$E\;=\;\Bigl\{\lambda=\sum_{i\in\mathcal V}\delta_{(i,x_i)}:\mathcal V\subset\mathcal I_c\text{ finite},\;x_i\in\mathbb R^d\Bigr\},$$17, a hybrid “pushmi-pullyu” behavior is observed: the mode stays near $$E\;=\;\Bigl\{\lambda=\sum_{i\in\mathcal V}\delta_{(i,x_i)}:\mathcal V\subset\mathcal I_c\text{ finite},\;x_i\in\mathbb R^d\Bigr\},$$18, but the mean drifts out, in agreement with the Bramson correction exponent $$E\;=\;\Bigl\{\lambda=\sum_{i\in\mathcal V}\delta_{(i,x_i)}:\mathcal V\subset\mathcal I_c\text{ finite},\;x_i\in\mathbb R^d\Bigr\},$$19 (Demircigil et al., 13 May 2025).

A related rank-based branching-selection system also converges to a reaction-diffusion equation. In the $$E\;=\;\Bigl\{\lambda=\sum_{i\in\mathcal V}\delta_{(i,x_i)}:\mathcal V\subset\mathcal I_c\text{ finite},\;x_i\in\mathbb R^d\Bigr\},$$20-BBM family, the hydrodynamic limit is

$$E\;=\;\Bigl\{\lambda=\sum_{i\in\mathcal V}\delta_{(i,x_i)}:\mathcal V\subset\mathcal I_c\text{ finite},\;x_i\in\mathbb R^d\Bigr\},$$21

and under additional assumptions the particle system has an asymptotic velocity

$$E\;=\;\Bigl\{\lambda=\sum_{i\in\mathcal V}\delta_{(i,x_i)}:\mathcal V\subset\mathcal I_c\text{ finite},\;x_i\in\mathbb R^d\Bigr\},$$22

The minimal spreading speed of the PDE is exactly $$E\;=\;\Bigl\{\lambda=\sum_{i\in\mathcal V}\delta_{(i,x_i)}:\mathcal V\subset\mathcal I_c\text{ finite},\;x_i\in\mathbb R^d\Bigr\},$$23, and $$E\;=\;\Bigl\{\lambda=\sum_{i\in\mathcal V}\delta_{(i,x_i)}:\mathcal V\subset\mathcal I_c\text{ finite},\;x_i\in\mathbb R^d\Bigr\},$$24, which is described there as a partial weak-selection principle (Mercer, 6 May 2026).

5. Long-time behavior, coexistence, extinction, and regulation

In mutually catalytic branching with negative correlations, the long-time behavior is governed by the transience or recurrence of the migration chain. For each fixed $$E\;=\;\Bigl\{\lambda=\sum_{i\in\mathcal V}\delta_{(i,x_i)}:\mathcal V\subset\mathcal I_c\text{ finite},\;x_i\in\mathbb R^d\Bigr\},$$25 and $$E\;=\;\Bigl\{\lambda=\sum_{i\in\mathcal V}\delta_{(i,x_i)}:\mathcal V\subset\mathcal I_c\text{ finite},\;x_i\in\mathbb R^d\Bigr\},$$26, coexistence of the two types occurs if and only if the underlying migration chain is transient; equivalently, if

$$E\;=\;\Bigl\{\lambda=\sum_{i\in\mathcal V}\delta_{(i,x_i)}:\mathcal V\subset\mathcal I_c\text{ finite},\;x_i\in\mathbb R^d\Bigr\},$$27

Otherwise, in the recurrent case,

$$E\;=\;\Bigl\{\lambda=\sum_{i\in\mathcal V}\delta_{(i,x_i)}:\mathcal V\subset\mathcal I_c\text{ finite},\;x_i\in\mathbb R^d\Bigr\},$$28

In particular, on $$E\;=\;\Bigl\{\lambda=\sum_{i\in\mathcal V}\delta_{(i,x_i)}:\mathcal V\subset\mathcal I_c\text{ finite},\;x_i\in\mathbb R^d\Bigr\},$$29, coexistence occurs precisely for $$E\;=\;\Bigl\{\lambda=\sum_{i\in\mathcal V}\delta_{(i,x_i)}:\mathcal V\subset\mathcal I_c\text{ finite},\;x_i\in\mathbb R^d\Bigr\},$$30 (Doering et al., 2011).

The discrete particle system with mutually catalytic interactions on $$E\;=\;\Bigl\{\lambda=\sum_{i\in\mathcal V}\delta_{(i,x_i)}:\mathcal V\subset\mathcal I_c\text{ finite},\;x_i\in\mathbb R^d\Bigr\},$$31 has the same phase boundary. For nearest-neighbor random walk, $$E\;=\;\Bigl\{\lambda=\sum_{i\in\mathcal V}\delta_{(i,x_i)}:\mathcal V\subset\mathcal I_c\text{ finite},\;x_i\in\mathbb R^d\Bigr\},$$32 implies coexistence possible, while $$E\;=\;\Bigl\{\lambda=\sum_{i\in\mathcal V}\delta_{(i,x_i)}:\mathcal V\subset\mathcal I_c\text{ finite},\;x_i\in\mathbb R^d\Bigr\},$$33 implies coexistence impossible (Fugenfirov et al., 2023). On finite tori, the empirical total masses converge after renormalization to the mutually catalytic diffusion

$$E\;=\;\Bigl\{\lambda=\sum_{i\in\mathcal V}\delta_{(i,x_i)}:\mathcal V\subset\mathcal I_c\text{ finite},\;x_i\in\mathbb R^d\Bigr\},$$34

in dimensions $$E\;=\;\Bigl\{\lambda=\sum_{i\in\mathcal V}\delta_{(i,x_i)}:\mathcal V\subset\mathcal I_c\text{ finite},\;x_i\in\mathbb R^d\Bigr\},$$35, under the transience bound $$E\;=\;\Bigl\{\lambda=\sum_{i\in\mathcal V}\delta_{(i,x_i)}:\mathcal V\subset\mathcal I_c\text{ finite},\;x_i\in\mathbb R^d\Bigr\},$$36 (Fugenfirov et al., 2023).

In branching diffusion with Ornstein–Uhlenbeck drift and pairwise attraction or repulsion, the center of mass decouples from the interaction parameter $$E\;=\;\Bigl\{\lambda=\sum_{i\in\mathcal V}\delta_{(i,x_i)}:\mathcal V\subset\mathcal I_c\text{ finite},\;x_i\in\mathbb R^d\Bigr\},$$37. Writing

$$E\;=\;\Bigl\{\lambda=\sum_{i\in\mathcal V}\delta_{(i,x_i)}:\mathcal V\subset\mathcal I_c\text{ finite},\;x_i\in\mathbb R^d\Bigr\},$$38

one obtains

$$E\;=\;\Bigl\{\lambda=\sum_{i\in\mathcal V}\delta_{(i,x_i)}:\mathcal V\subset\mathcal I_c\text{ finite},\;x_i\in\mathbb R^d\Bigr\},$$39

Hence, almost surely, $$E\;=\;\Bigl\{\lambda=\sum_{i\in\mathcal V}\delta_{(i,x_i)}:\mathcal V\subset\mathcal I_c\text{ finite},\;x_i\in\mathbb R^d\Bigr\},$$40 when $$E\;=\;\Bigl\{\lambda=\sum_{i\in\mathcal V}\delta_{(i,x_i)}:\mathcal V\subset\mathcal I_c\text{ finite},\;x_i\in\mathbb R^d\Bigr\},$$41, whereas $$E\;=\;\Bigl\{\lambda=\sum_{i\in\mathcal V}\delta_{(i,x_i)}:\mathcal V\subset\mathcal I_c\text{ finite},\;x_i\in\mathbb R^d\Bigr\},$$42 when $$E\;=\;\Bigl\{\lambda=\sum_{i\in\mathcal V}\delta_{(i,x_i)}:\mathcal V\subset\mathcal I_c\text{ finite},\;x_i\in\mathbb R^d\Bigr\},$$43, with $$E\;=\;\Bigl\{\lambda=\sum_{i\in\mathcal V}\delta_{(i,x_i)}:\mathcal V\subset\mathcal I_c\text{ finite},\;x_i\in\mathbb R^d\Bigr\},$$44 Gaussian (Englander et al., 2016). For the relative system $$E\;=\;\Bigl\{\lambda=\sum_{i\in\mathcal V}\delta_{(i,x_i)}:\mathcal V\subset\mathcal I_c\text{ finite},\;x_i\in\mathbb R^d\Bigr\},$$45, if $$E\;=\;\Bigl\{\lambda=\sum_{i\in\mathcal V}\delta_{(i,x_i)}:\mathcal V\subset\mathcal I_c\text{ finite},\;x_i\in\mathbb R^d\Bigr\},$$46 then the empirical measure converges almost surely to

$$E\;=\;\Bigl\{\lambda=\sum_{i\in\mathcal V}\delta_{(i,x_i)}:\mathcal V\subset\mathcal I_c\text{ finite},\;x_i\in\mathbb R^d\Bigr\},$$47

while for $$E\;=\;\Bigl\{\lambda=\sum_{i\in\mathcal V}\delta_{(i,x_i)}:\mathcal V\subset\mathcal I_c\text{ finite},\;x_i\in\mathbb R^d\Bigr\},$$48 local extinction always occurs: every fixed bounded set is eventually empty (Englander et al., 2016).

Self-regulation and competition replace coexistence thresholds by carrying-capacity effects. In the multi-type spatial branching model for local self-regulation, the diffusion limit has local logistic drift $$E\;=\;\Bigl\{\lambda=\sum_{i\in\mathcal V}\delta_{(i,x_i)}:\mathcal V\subset\mathcal I_c\text{ finite},\;x_i\in\mathbb R^d\Bigr\},$$49, and the abstract states that the densities for large times do not depend on the initial distribution but mainly on the carrying capacities (Greven et al., 2015). In the one-dimensional competition model,

$$E\;=\;\Bigl\{\lambda=\sum_{i\in\mathcal V}\delta_{(i,x_i)}:\mathcal V\subset\mathcal I_c\text{ finite},\;x_i\in\mathbb R^d\Bigr\},$$50

subcriticality and extinction are characterized by

$$E\;=\;\Bigl\{\lambda=\sum_{i\in\mathcal V}\delta_{(i,x_i)}:\mathcal V\subset\mathcal I_c\text{ finite},\;x_i\in\mathbb R^d\Bigr\},$$51

(Ba et al., 2013).

These results show that interacting branching diffusions exhibit several distinct long-time regimes: coexistence driven by transience, extinction driven by recurrence or outward drift, regulation by carrying capacities, and front propagation governed by minimal-wave-speed selection.

6. Duality, spine methods, control, and structural representations

Duality is a principal tool when interaction destroys the classical branching property. For negatively correlated mutually catalytic branching, the exact mixed second-moment formula is

$$E\;=\;\Bigl\{\lambda=\sum_{i\in\mathcal V}\delta_{(i,x_i)}:\mathcal V\subset\mathcal I_c\text{ finite},\;x_i\in\mathbb R^d\Bigr\},$$52

where $$E\;=\;\Bigl\{\lambda=\sum_{i\in\mathcal V}\delta_{(i,x_i)}:\mathcal V\subset\mathcal I_c\text{ finite},\;x_i\in\mathbb R^d\Bigr\},$$53 is the collision local time of two independent migration chains (Doering et al., 2011). Since $$E\;=\;\Bigl\{\lambda=\sum_{i\in\mathcal V}\delta_{(i,x_i)}:\mathcal V\subset\mathcal I_c\text{ finite},\;x_i\in\mathbb R^d\Bigr\},$$54, the factor $$E\;=\;\Bigl\{\lambda=\sum_{i\in\mathcal V}\delta_{(i,x_i)}:\mathcal V\subset\mathcal I_c\text{ finite},\;x_i\in\mathbb R^d\Bigr\},$$55 decays on collisions, which is the key mechanism behind extinction in the recurrent case (Doering et al., 2011). The same paper combines this with exit-time estimates for correlated Brownian motion in the first quadrant and uniform integrability bounds independent of the branching rate, allowing passage to the infinite-rate limit (Doering et al., 2011).

Exponential duality also appears in multitype self-regulating systems. In the exchangeable case, Greven–Sturm–Winter–Zähle construct a dual process $$E\;=\;\Bigl\{\lambda=\sum_{i\in\mathcal V}\delta_{(i,x_i)}:\mathcal V\subset\mathcal I_c\text{ finite},\;x_i\in\mathbb R^d\Bigr\},$$56 and the duality function

$$E\;=\;\Bigl\{\lambda=\sum_{i\in\mathcal V}\delta_{(i,x_i)}:\mathcal V\subset\mathcal I_c\text{ finite},\;x_i\in\mathbb R^d\Bigr\},$$57

which yields uniqueness of the exchangeable diffusion martingale problem and supports equilibrium and coexistence analysis (Greven et al., 2015). For mutually catalytic branching particle systems on finite tori, approximate self-duality based on

$$E\;=\;\Bigl\{\lambda=\sum_{i\in\mathcal V}\delta_{(i,x_i)}:\mathcal V\subset\mathcal I_c\text{ finite},\;x_i\in\mathbb R^d\Bigr\},$$58

is central in identifying the finite-system-scheme limit (Fugenfirov et al., 2023).

Spine constructions provide another structural method. For continuous type-space branching processes with interactions, Medous introduces a Girsanov-type change of measure that singles out a distinguished lineage and proves the many-to-one identity

$$E\;=\;\Bigl\{\lambda=\sum_{i\in\mathcal V}\delta_{(i,x_i)}:\mathcal V\subset\mathcal I_c\text{ finite},\;x_i\in\mathbb R^d\Bigr\},$$59

with path-integral weight $$E\;=\;\Bigl\{\lambda=\sum_{i\in\mathcal V}\delta_{(i,x_i)}:\mathcal V\subset\mathcal I_c\text{ finite},\;x_i\in\mathbb R^d\Bigr\},$$60 determined by the nonlinear operator $$E\;=\;\Bigl\{\lambda=\sum_{i\in\mathcal V}\delta_{(i,x_i)}:\mathcal V\subset\mathcal I_c\text{ finite},\;x_i\in\mathbb R^d\Bigr\},$$61 (Medous, 2023). If $$E\;=\;\Bigl\{\lambda=\sum_{i\in\mathcal V}\delta_{(i,x_i)}:\mathcal V\subset\mathcal I_c\text{ finite},\;x_i\in\mathbb R^d\Bigr\},$$62, then $$E\;=\;\Bigl\{\lambda=\sum_{i\in\mathcal V}\delta_{(i,x_i)}:\mathcal V\subset\mathcal I_c\text{ finite},\;x_i\in\mathbb R^d\Bigr\},$$63, recovering a classical many-to-one formula. The same framework yields a generalized continuous-time Kesten–Stigum theorem with interactions and an alternative spine construction for exact simulation of stochastic size-dependent populations (Medous, 2023).

A complementary structural representation is the generalized Ray–Knight theorem for branching with competition. Under Hypothesis B, the reflected Brownian motion

$$E\;=\;\Bigl\{\lambda=\sum_{i\in\mathcal V}\delta_{(i,x_i)}:\mathcal V\subset\mathcal I_c\text{ finite},\;x_i\in\mathbb R^d\Bigr\},$$64

has local times whose inverse-local-time slices encode the diffusion $$E\;=\;\Bigl\{\lambda=\sum_{i\in\mathcal V}\delta_{(i,x_i)}:\mathcal V\subset\mathcal I_c\text{ finite},\;x_i\in\mathbb R^d\Bigr\},$$65, and for $$E\;=\;\Bigl\{\lambda=\sum_{i\in\mathcal V}\delta_{(i,x_i)}:\mathcal V\subset\mathcal I_c\text{ finite},\;x_i\in\mathbb R^d\Bigr\},$$66,

$$E\;=\;\Bigl\{\lambda=\sum_{i\in\mathcal V}\delta_{(i,x_i)}:\mathcal V\subset\mathcal I_c\text{ finite},\;x_i\in\mathbb R^d\Bigr\},$$67

This identifies the full branching diffusion field with the local-time profile of a single reflected Brownian motion whose drift depends on accumulated local time through $$E\;=\;\Bigl\{\lambda=\sum_{i\in\mathcal V}\delta_{(i,x_i)}:\mathcal V\subset\mathcal I_c\text{ finite},\;x_i\in\mathbb R^d\Bigr\},$$68 (Ba et al., 2013).

Optimal control adds a PDE layer to interacting branching diffusions. In the controlled setting, the dynamic programming principle yields an infinite system of coupled Hamilton–Jacobi–Bellman equations on the disjoint union $$E\;=\;\Bigl\{\lambda=\sum_{i\in\mathcal V}\delta_{(i,x_i)}:\mathcal V\subset\mathcal I_c\text{ finite},\;x_i\in\mathbb R^d\Bigr\},$$69, with value functions

$$E\;=\;\Bigl\{\lambda=\sum_{i\in\mathcal V}\delta_{(i,x_i)}:\mathcal V\subset\mathcal I_c\text{ finite},\;x_i\in\mathbb R^d\Bigr\},$$70

Under coercivity, continuity and growth bounds transfer to the value function, which is the unique continuous viscosity solution in the corresponding growth class; a comparison principle is proved by a doubling-of-variables argument with a penalization in $$E\;=\;\Bigl\{\lambda=\sum_{i\in\mathcal V}\delta_{(i,x_i)}:\mathcal V\subset\mathcal I_c\text{ finite},\;x_i\in\mathbb R^d\Bigr\},$$71 (Ocello, 16 Jan 2026). In the permutation-invariant mean-field regime, optimal controls can be chosen symmetric, and the HJB system reduces to a single PDE on measure space, written in the paper as a master-equation involving non-local derivatives $$E\;=\;\Bigl\{\lambda=\sum_{i\in\mathcal V}\delta_{(i,x_i)}:\mathcal V\subset\mathcal I_c\text{ finite},\;x_i\in\mathbb R^d\Bigr\},$$72 (Ocello, 16 Jan 2026).

Taken together, these methods show that branching diffusion with interactions is not a single model but a framework. Its state spaces are measure-valued and genealogical, its nonlinearities may be rank-based, catalytic, mean-field, or competitive, and its analysis proceeds through martingale problems, hydrodynamic limits, scaling limits, traveling-wave theory, duality, spinal decompositions, and viscosity solutions.