Inhomogeneous Immigration Processes

• Inhomogeneous immigration processes are stochastic models that introduce new elements into a system at variable rates, significantly impacting dynamics and critical thresholds.
• They employ frameworks like shot noise and nonhomogeneous Poisson and determinantal point processes to capture time- and space-dependent immigration events.
• Analytical techniques including martingale methods, operator theory, and functional limit theorems reveal diverse asymptotic behaviors with applications in ecology, genetics, and network systems.

Inhomogeneous immigration processes are stochastic models in which new elements (particles, individuals, agents, or “immigrants”) are introduced into a system at rates or times that vary in space or time or depend on additional random mechanisms. These processes arise in a variety of branches including population dynamics, fragmentation, branching structures, random functionals, queueing, and statistical physics. The inhomogeneity specifically refers to the time-dependence, spatial-dependence, randomness, or correlation structure of the immigration mechanism, which may interact non-trivially with intrinsic system dynamics (e.g., reproduction, fragmentation, dispersal) and underlying environment.

1. Mathematical Formulations and Modeling Frameworks

A wide spectrum of frameworks captures inhomogeneous immigration. A prototypical construction is via shot noise or superposition processes, in which immigrants arrive at times $\{T_k\}$, and each immigrant is associated with a (possibly random) evolution $X_k$ (typically in Skorokhod space $D(\mathbb{R})$):

$$Y(t) = \sum_{k \ge 0} X_{k+1}(t - T_k) 1_{\{T_k \leq t\}}$$

In classical models, $\{T_k\}$ often form a renewal process with fixed interarrival law; for inhomogeneous immigration, $\{T_k\}$ may be arbitrary, governed by nonhomogeneous Poisson processes, determinantal point processes, Yule processes, or even random times from more general point process structures (Marynych, 2015, Dong et al., 2019, Minchev et al., 19 Nov 2024, Tung et al., 19 Aug 2025).

Key types of inhomogeneity:

• Time-inhomogeneous rates: Immigration rates $\lambda(t)$ vary in time, e.g., Poisson process with intensity $r(t)$ converging to a constant or with prescribed nontrivial temporal evolution (Mitov et al., 4 Jan 2025, Tung et al., 19 Aug 2025).
• Immigration with random environments: Immigrants arrive according to rates or laws modulated by random environments, either stationary or nonstationary (Han et al., 2018).
• Correlated immigration times: Use of point processes beyond the Poisson, notably determinantal point processes (DPP), induces negative correlations (“repulsion”) between immigration times (Minchev et al., 19 Nov 2024).
• Generation-dependent immigration: In branching processes, immigration law $h_n(s)$ varies with generation $n$ (González et al., 28 Jan 2024).
• Multi-type and spatial extension: High-dimensional models allow for vector-valued immigration mechanisms, spatial migration, or type-specific inhomogeneity (Györfi et al., 2013, Li et al., 2015, Minchev et al., 19 Nov 2024).

2. Limit Theorems and Asymptotics

Asymptotic distributional results for inhomogeneous immigration processes depend on both the arrival mechanism and the properties of the system’s intrinsic dynamics. Several key types of limit behavior are established in the literature:

• Gamma limits in critical branching with stabilized inhomogeneous immigration: For BPVEI (branching processes in varying environment with generation-dependent immigration) with critical offspring distributions and immigration means stabilizing to a positive value, the normalized process converges in law to a gamma distribution (González et al., 28 Jan 2024).
• Poisson and negative binomial limits for Galton–Watson processes: In nearly critical inhomogeneous GW processes, Poisson or compound Poisson behavior emerges when the offspring variance vanishes quickly compared to the “distance from criticality,” while a negative binomial limit arises when the offspring variance is not negligible (Kevei, 2011).
• Stable and extremal limits in heavy-tailed immigration: For continuous-state branching processes with large/“super-log” immigration tails (i.e., no log-moment for the immigration measure), scaling limits are non-classical: the process may converge to subordinators, continuous-state branching processes, extremal processes, or extremal shot noise (ESN) processes. If the immigration is exceedingly heavy-tailed, a supremum-type ESN limit occurs, dominated by the rare immigration events (Foucart et al., 2023).
• Extreme value theory for passage times: The statistics of the fastest arrivals (first passage times) in time-inhomogeneous or Yule-type immigration processes deviate from the classical Frechet, Gumbel, Weibull families due to non-iid, correlated arrival and search times. The Yule process, in particular, yields a logistic limit law for the fastest arrival, i.e., a “difference of two independent Gumbel random variables,” reflecting the combined effect of exponential growth and stochastic timing (Tung et al., 19 Aug 2025).

A summary table:

Regime	Normalization	Limit Law	Context
Critical BPVEI w/ gen-dep. immigration	$a_n$	Gamma$(2\alpha/\nu,1)$	(González et al., 28 Jan 2024)
GW w/ inhomogeneous immigration	Poisson scaling	Poisson or negative binomial	(Kevei, 2011, Györfi et al., 2013)
Infinite variance/reproduction	$\sim t^{-1/\beta}$	Stable/ESN/Uniform	(Mitov et al., 4 Jan 2025, Foucart et al., 2023)
Fast TII / Yule immigration (FPT)	$a_\lambda$, $b_\lambda$	Incomplete gamma / logistic	(Tung et al., 19 Aug 2025)

3. Effects of Inhomogeneity on Population-Level Evolution

Inhomogeneous immigration can dramatically influence the qualitative and quantitative behavior of populations, fragmentation, and genealogies:

• Extinction and persistence criteria: In branching processes, extinction probabilities and the existence of nontrivial limiting distributions are intertwined with the asymptotic behavior and moments of the generation-dependent immigration law. Explicit integral conditions characterize almost sure extinction (González et al., 28 Jan 2024, Li et al., 2015, Mitov et al., 4 Jan 2025). In Markov branching processes with infinite mean immigration, extinction may be precluded altogether, or distinct regimes (with stable, uniform, or degenerate limits) can occur.
• Spatial heterogeneity and pattern formation: In models of infinite populations in continuous space, spatially inhomogeneous immigration rates $b(x)$ can give rise to spatial diversity and nontrivial patterning, modulated by local particle repulsion or habitat structure (Kozitsky, 2018). The interplay of the immigration field and spatial interactions dictates whether the system maintains spatial heterogeneity or converges to homogeneity.
• Network models and social dynamics: In agent-based emigration/immigration models on networked populations, the combination of local social influences and economic (pecuniary) incentives introduces variability in migration patterns that reflects both micro-level (personal) and macro-level (national wealth, density) heterogeneities. Models on small-world networks demonstrate cascades, threshold effects, and equilibria shaped by local and global parameters (Fotouhi et al., 2012).

4. Analytical and Probabilistic Techniques

Rigorous characterization of inhomogeneous immigration processes leverages several classes of tools:

• Martingale and submartingale techniques: Used extensively for limit theorems, estimation (CLS estimators), and convergence proofs in contexts such as fragmentation with immigration and branching processes (Knobloch, 2012, Rahimov et al., 2012).
• Operator and Lie-algebraic methods: Time-inhomogeneous birth–death–immigration processes require analyzing the closures of operator algebras, and restrictions on the birth-immigration rate relation ($y(t)=B X(t)$) are necessary for obtaining tractable, closed-form expressions via the Wei–Norman method (Ohkubo, 2014).
• Functional limit theorems in path spaces: Results are often formulated as weak convergence in Skorokhod space $D(\mathbb{R})$ equipped with the $J_1$-topology, both for random process convergence to stationarity and for shot noise functionals with general arrival and response processes (Iksanov et al., 2013, Marynych, 2015, Dong et al., 2019).
• Point process and Laplace functional methods: Immigration mechanisms driven by point processes (Poisson, Cox, DPP) enable explicit Laplace-transforms/generating functions for the resultant branching or multi-type processes, accommodating inhomogeneous or correlated arrivals (Minchev et al., 19 Nov 2024).
• Renewal-theoretic arguments and Toeplitz-type summations: Applied for asymptotics in the context of GW and INAR-type models with inhomogeneous immigration (Kevei, 2011, Györfi et al., 2013, González et al., 28 Jan 2024).

5. Cross-disciplinary Applications

The models and results from inhomogeneous immigration processes are applicable in:

• Population genetics and evolutionary biology: The theory of distinguished exchangeable coalescents (M-coalescents) with genealogical “immigrant blocks” connects directly to models for populations with strong migration or seed-bank effects (Foucart, 2010).
• Epidemiology and ecology: Immigration terms model influx of individuals, infections, or recoveries into subpopulations, with inhomogeneity capturing variable contact or migration patterns.
• Queueing and service systems: The GI/G/$\infty$ queue with general arrival processes maps directly to functional limit theorems for immigration shot noise (Marynych, 2015).
• Statistical physics and spatial ecology: Infinite particle systems with spatially heterogeneous immigration and local interactions are relevant for habitat colonization, clustering, and pattern formation (Kozitsky, 2018).
• Branching random walks, network protocols, and risk processes: Nonhomogeneous immigrant arrivals (as in inhomogeneous Poisson or DPP immigration processes) capture correlated “shock” events, patchy resource inflow, and irregular jumps in evolving network structures (Minchev et al., 19 Nov 2024).

6. Notable Limitations and Open Directions

Several technical and modeling limitations are highlighted in the literature:

• Exact analytic solutions for time-inhomogeneous immigration processes are generally intractable unless structural restrictions (e.g., proportionality between birth and immigration rates) yield a finite-dimensional Lie algebra (Ohkubo, 2014).
• Functional limit results for general point-process-driven immigration, including renewal and non-renewal shot noise, depend crucially on integrability and mixing conditions; further, extensions to dependencies, clustering, or alternative topologies are nontrivial (Iksanov et al., 2013, Marynych, 2015).
• Extreme value statistics in processes with strongly correlated or non-i.i.d. passage times provide non-classical and non-universal limits, deviating from traditional extreme value theory (Tung et al., 19 Aug 2025).
• In the presence of both heavy-tailed offspring and immigration (infinite variance and/or mean), classification of limit laws and survival-type probabilities remains sensitive to the precise balance of parameters and slowly varying functions, necessitating delicate asymptotic analysis (Mitov et al., 4 Jan 2025, Foucart et al., 2023).

7. Summary Table of Key Regimes

Model Class	Immigration Inhomogeneity	Limiting Behavior/Result	Primary References
Branching w/ gen.-dep. immigration (BPVEI)	Time/generation inhomogeneous	Extinction criteria and gamma limits	(González et al., 28 Jan 2024)
Multi-type GW, compound Poisson limit	Type/generation inhomogeneous	Dependent coordinates in distribution	(Györfi et al., 2013)
Random processes, general arrival times	Arbitrary/random pulse times	Gaussian processes via wide-sense regularity	(Dong et al., 2019)
Infinite pop. in spatial habitat	Spatially varying immigration	Mesoscopic-microscopic kinetic agreement	(Kozitsky, 2018)
Fast inhomogeneous immigration (extreme values)	TII/Yule/time-inhomogeneous	Logistic/extremal (non-standard) limits	(Tung et al., 19 Aug 2025)
DPP immigration in multi-type processes	Correlated/repulsive arrivals	Negative covariance, Laplace transform method	(Minchev et al., 19 Nov 2024)

8. Concluding Perspective

Inhomogeneous immigration processes form a central and rich theme connecting the structure and evolution of stochastic systems ranging from population models and fragmentation to spatial and networked systems. The interplay between system dynamics and the inhomogeneous, possibly correlated or heavy-tailed nature of immigration profoundly influences extinction, persistence, correlation structure, and macroscopic properties. Modern analysis combines probabilistic, operator-theoretic, and functional analytic techniques to capture both universal regimes and model-specific nuances, with practical implications in ecology, genetics, statistical physics, and beyond. The current research identifies both general principles—such as critical parameter regimes—and highlights the particularities introduced by correlation, heavy tails, and temporal or spatial structure in the immigration mechanism.