CIMBI Processes: Multitype Branching with Immigration

• CIMBI processes are continuous-state Markov models that extend classical CBI frameworks by incorporating quadratic inter-type interactions in multitype populations.
• They employ stochastic differential equations with both diffusion and jump components to simultaneously capture branching, immigration, and competitive or cooperative dynamics.
• This framework aids in analyzing long-term behaviors, boundary non-attainment, and moment recursions, with applications in biology, epidemiology, and network science.

A continuous-state interacting multi-type branching process with immigration (CIMBI process) is a Markov process on $\mathbb{R}_+^d$ modeling the stochastic evolution of $d$ population types undergoing branching, immigration, interaction, and random fluctuations. Interactions may be competitive, cooperative, or mixed, are typically quadratic in the population masses, and the population evolves according to a system of stochastic differential equations (SDEs) with both diffusion and jump components. The CIMBI framework unifies and extends classical continuous-state branching processes with immigration (CBI) by allowing for nontrivial inter-type interactions, providing a broad and flexible class for modeling multitype populations in biology, epidemiology, and related fields.

1. Mathematical Structure and Definition

Let $d\geq1$. The canonical state space is $\mathbb{R}_+^d$. The process $X=(X_1,\ldots,X_d)$ is defined, under admissible parameters, by the following SDE with jumps: $$\begin{aligned} X_i(t) &= x_i + \int_0^t \left[\, \eta_i + \sum_{j=1}^d b_{ij}X_j(s) + \gamma_i(X(s))\,\right] ds + \int_0^t \sqrt{2\sigma_i X_i(s)}\,dW_i(s) \ &\quad+ \int_0^t \int_{\mathbb{R}_+^d} z_i\,N_0(ds,dz) + \int_0^t \int_0^\infty \int_{\mathbb{R}_+^d} z_i\,\mathbf{1}_{\{u\leq X_i(s-)\}}\,\tilde N_i(ds,du,dz) \ &\quad+ \sum_{j\ne i} \int_0^t \int_0^\infty \int_{\mathbb{R}_+^d} z_i\,\mathbf{1}_{\{u\le X_j(s-)\}}\,N_j(ds,du,dz), \end{aligned}$$ for $i=1,\ldots,d$. Here:

• $\eta_i$ is the type-$i$ immigration rate,
• $\sigma_i > 0$ is the diffusion coefficient,
• $B=(b_{ij})$ governs linear drift and cross-type reproduction,
• $C=(c_{ij})$ encodes quadratic interaction, where $\gamma_i(x)=\sum_{j=1}^d c_{ij} x_i x_j$,
• $N_0$ and $N_i$ are independent Poisson random measures governing immigration and branching jumps,
• $\tilde N_i$ is the compensated Poisson measure,
• $W=(W_1,\ldots,W_d)$ is standard $d$-dimensional Brownian motion.

Admissibility requires (i) $b_{ij}\geq 0$ for $i\ne j$, (ii) $c_{ii}<0$ for all $i$ and (iii) $\sum_{i,j}c_{ij} x_i x_j \geq 0$ for all $x\in\mathbb{R}_+^d$ to guarantee non-explosion and well-posedness (Jin et al., 24 Dec 2025).

The infinitesimal generator $L$ of a non-interacting multi-type CBI process ($C=0$) is

$$\begin{aligned} L f(x) &= \sum_{j=1}^d c_j x_j \frac{\partial^2 f}{\partial x_j^2}(x) + \langle \beta + Bx, \nabla f(x) \rangle \ & \qquad + \int_{\mathbb{R}_+^d}[f(x + z) - f(x)]\,\nu(dz) \ & \qquad + \sum_{j=1}^d x_j \int_{\mathbb{R}_+^d} [f(x+z) - f(x) - \langle z,\nabla f(x)\rangle] \mu_j(dz), \end{aligned}$$

with admissible diffusion, drift, and Lévy measures (Barczy et al., 2014).

2. Branching, Immigration, and Interaction Mechanisms

The process is characterized by vector-valued branching and scalar immigration mechanisms, each of Lévy-Khintchine type:

• Immigration mechanism: $$F(\xi) = \langle\beta, \xi\rangle + \int (1 - e^{-\langle\xi,z\rangle})\,\nu(dz)$$
• Branching mechanism for type $j$:

$$R_j(\xi) = c_j \xi_j^2 - \langle B e_j, \xi\rangle + \int \left(e^{-\langle\xi,z\rangle} - 1 + \langle \xi, z \rangle\right) \mu_j(dz)$$

• Interaction mechanism: Quadratic terms $\gamma_i(x)$ modifying the drift, typically with $c_{ii}<0$ (self-competition) but arbitrary cross-terms (Jin et al., 24 Dec 2025).

These mechanisms yield an affine Laplace transform for $X_t$: $$\mathbb{E}\left[e^{-\langle\xi,X_t\rangle}\right] = \exp\Bigg(-\langle x, v(t,\xi)\rangle - \int_0^t F(v(s,\xi))\,ds\Bigg)$$ where $v(t,\xi)$ solves a generalized Riccati ODE driven by $R(\cdot)$ (Friesen et al., 2019).

3. Jump Structure and Path Properties

Jumps in CIMBI processes originate from both branching and immigration, with detailed jump law as follows (Barczy et al., 2023):

• The total Lévy measure for jumps in $A \subset \mathbb{R}_+^d \setminus\{0\}$ is $\Pi(A) = \nu(A) + \sum_i\mu_i(A)$.
• The process of jumps of size in $A$ is a Cox process with random intensity $\lambda_t = \nu(A) + \sum_i X_{t-,i} \mu_i(A)$.
• The probability that no jump lands in $A$ until time $t$ is conditionally

$$\mathbb{P}(T_A > t \mid X_s, s \le t) = \exp\left(-\int_0^t [\nu(A) + \sum_i X_{s,i} \mu_i(A)] ds\right).$$

• If $A$ is a non-degenerate rectangle anchored at zero and $\Pi(A)=0$, then almost surely no jump ever lands in $A$; conversely, $$\mathbb{P}(\sup_{s\le t} \Delta X_s \in A) = 0 \Leftrightarrow \Pi(A)=0$$ (Barczy et al., 2023).

The overall process exhibits both continuous paths (from Feller diffusion) and compound Poisson jumps, with pathwise non-extinction/transience dictated by parameter integrals (Friesen et al., 2019).

4. Boundary Attainment, Non-Extinction, and Persistence

Boundary behavior in interacting multi-type branching processes with immigration reveals regimes of permanent coexistence or unavoidable extinction (Jin et al., 24 Dec 2025, Friesen et al., 2019):

• Non-attainment (persistence): If $\eta_i > \sigma_i$ for every $i$, or $\eta_i = \sigma_i$ and the small-jump measure $\mu_i$ is integrable near zero, then (with positive initial masses and interaction satisfying $\sum_{i,j}c_{ij}x_i x_j\ge0$) the process remains strictly inside the positive orthant with probability one: $\mathbb{P}[X_i(t) > 0,\ \forall t] = 1$.
• Extinction (boundary hitting): If immigration is weak ($\eta_i \le \frac12 \sigma_i$) and self-competition or damping is strong, then pure-diffusion CIMBI processes almost surely hit the boundary in finite time; with finite jump activity, hitting occurs with strictly positive probability.
• Mechanistic interpretation: Immigration acts as rescue; competition via $C$ drives unstable types to extinction, while cooperation or sufficient immigration can enable coexistence (Jin et al., 24 Dec 2025).

A general sufficient condition for non-extinction is the divergence of the integral

$$\int_{\kappa}^\infty \exp\left(\int_{\kappa}^\xi \frac{F^{(k)}(u)}{R^{(k)}(u)} du\right)\frac{1}{R^{(k)}(\xi)}\,d\xi = \infty \implies \mathbb{P}[X_k(t)>0,\ \forall t]=1,$$

for each coordinate (Friesen et al., 2019).

5. Asymptotic and Ergodic Behavior

Long-time growth, ergodicity, and limiting shape are governed by the spectral properties of the linear component $B$ and immigration/branching moments (Barczy et al., 2018, Barczy et al., 2018, Barczy et al., 2014, Chen et al., 2021):

• Supercritical regime: If the Perron root $s(B)>0$, $e^{-s t} X(t)$ converges almost surely and in $L^1$ to $W u$ for a deterministic direction $u>0$; the random amplitude $W$ reflects both initial mass and cumulative immigration.
• Type-frequency convergence: Relative type frequencies converge almost surely to the Perron right-eigenvector $u$ under first-moment immigration assumptions.
• Fluctuations: For projections onto non-Perron spectral directions, central limit theorems provide Gaussian or mixed-normal fluctuation limits under appropriate higher moments.
• Critical scaling: For $s(B)=0$ (the critical case), a properly rescaled process converges to a squared-Bessel process on the Perron ray; population structure collapses to the leading eigendirection (Barczy et al., 2014).

Ergodicity in the presence of negative real parts for the drift matrix and suitable moment conditions yields exponential convergence in Wasserstein distance to the stationary law (Chen et al., 2021).

6. Moment Recursions, Densities, and Regularity

Explicit recursions for mixed and central moments of all orders are available and are polynomials in the initial state, with degree at most $k$ and $\lfloor k/2\rfloor$ for the $k$-th moment and central moment, respectively (Barczy et al., 2014). Transition densities exist and possess anisotropic Besov regularity under non-degenerate noise in each direction ($\alpha_i>4/3$ for all $i$), even in the absence of full diffusion (Friesen et al., 2018).

7. Biological and Theoretical Significance

CIMBI processes rigorously encode population systems with multiple interacting types, integrating continuous branching, instantaneous jumps, rich immigration, and broad inter-specific interactions. The possibility of boundary non-attainment enables mathematical representations of permanently coexisting populations; extinction criteria delineate when stochasticity and competition lead to loss of types. These models, with their explicit moment and density structure, serve as theoretical foundations for statistical inference, simulation, and deeper study of interacting stochastic population systems in ecology, epidemiology, and network science (Jin et al., 24 Dec 2025, Friesen et al., 2019, Barczy et al., 2018).