Supercritical Non-Local Superprocess

• Supercritical non-local superprocesses are measure-valued Markov processes featuring non-local branching and exponential mass growth governed by a principal eigenvalue.
• They exhibit law-of-large-numbers and central limit behaviors with fluctuations dependent on spectral gaps and normalization conditions.
• Backbone and skeleton decompositions provide structural insights into genealogical dynamics and enable precise analysis of limit and fluctuation theorems.

A supercritical non-local superprocess is a class of measure-valued Markov processes characterized by the following central properties: spatial motion driven by a Markov process (typically non-symmetric), a branching mechanism that allows offspring to be generated non-locally (offspring can be spatially displaced away from the parent location), and exponential growth in total mass as determined by the principal eigenvalue of the first moment semigroup. The formalism provides a theoretical foundation for branching systems with both spatial heterogeneity and general nonlocal reproduction, leading to rich fluctuation and limit behaviors that differ strongly from classical locally branching superprocesses. The recent literature provides a comprehensive description of law-of-large-numbers limits, spatial and functional central limit theorems, backbone and skeleton decompositions, regime-dependent fluctuation results, and precise almost sure fluctuation rates including laws of the iterated logarithm.

1. Mathematical Structure and Non-Local Branching

The state space is a Luzin space $E$. The process $X = \{X_t: t \geq 0\}$ evolves according to a spatial Markov process, with non-local branching mechanism

$$\Psi(x, f) = a(x) f(x) + b(x) f(x)^2 - n(x, f) + \big(e^{-\nu(f)} - 1 + \nu(\{x\}) f(x)\big) H(x, d\nu),$$

where $a(x)$ and $b(x)$ describe linear and quadratic local contributions, $n(x, f)$ is the contribution from local jumps, and $H(x, d\nu)$ encodes the non-local (offspring-dispersing) component. The process is called supercritical if the first moment semigroup $T_t$ has a Perron–Frobenius behavior: there exists $(\lambda_1, \varphi, \widetilde{\varphi})$, with $\lambda_1>0$, $T_t\varphi = e^{\lambda_1 t}\varphi$, and $$(T_t f, \widetilde{\varphi}) = e^{\lambda_1 t}(f, \widetilde{\varphi})$$ for all bounded measurable $f$.

Supercriticality implies exponential growth. The process admits a fundamental martingale

$$W_t = e^{-\lambda_1 t} \langle \varphi, X_t \rangle,$$

which converges almost surely to a non-degenerate limit $W_\infty$ under a second moment (or $L\log L$) condition, as shown in (Ren et al., 2017, Chen et al., 2017), and (Ren et al., 2018). If the $L\log L$ condition fails, proper normalization $\gamma_t$ yields a nontrivial limit for $\gamma_t\langle\varphi, X_t\rangle$ (Ren et al., 2017).

2. Law of Large Numbers, Spectral Theory, and Limit Theorems

The long-time asymptotic behavior of $X_t$ is governed by the principal eigentriple of the mean semigroup. Weak and strong laws of large numbers (LLNs and SLLNs) hold under general non-local branching mechanisms: $\lim_{t \to \infty} e^{-\lambda_1 t} \langle f, X_t \rangle = (f, \widetilde{\varphi}) W_\infty, \quad \text{in %%%%18%%%% and a.s. for bounded %%%%19%%%%}.$ Conditions for these theorems are detailed in (Palau et al., 2018, Chen et al., 2017), and include spectral gap, martingale, and moment criteria. The left-right eigenpair $(\varphi, \widetilde{\varphi})$ describes the spatial profile and determines large deviations (Yang, 23 Mar 2025).

The role of the first moment eigenstructure is central: if $T_t\varphi = e^{\lambda_1 t}\varphi$, scaling by $e^{-\lambda_1 t}$ “removes” the mean exponential growth, allowing convergence of $\langle f, X_t \rangle$ to be dominated by the projection on $\widetilde{\varphi}$.

3. Fluctuations, Central Limit Theorems, and Functional Fluctuation Regimes

Second-order behavior of linear functionals is intricate and governed by a spectral decay parameter $\epsilon(f)$. For bounded measurable $f$,

$$\epsilon(f) := \sup\left\{ r\in\mathbb{R} : \limsup_{t\to\infty} \frac{1}{t} \log \sup_{x} |p(x)^{-1} e^{-\lambda_1 t} T_t f(x) - (f, \widetilde{\varphi})| < -r \right\}$$

(Yang, 23 Mar 2025). Fluctuation regimes are as follows:

• If $\epsilon(f) > \lambda_1/2$, normalized fluctuations

$$\frac{e^{-\lambda_1 t} \langle f, X_t \rangle - (f, \widetilde{\varphi}) W_\infty}{e^{-\lambda_1 t/2}}$$

converge in distribution to a nondegenerate Gaussian random variable.

• If $\epsilon(f) = \lambda_1/2$, an additional polynomial-in-$t$ normalization is required; the limit remains Gaussian.
• If $\epsilon(f) < \lambda_1/2$, the central limit scaling is insufficient; a larger normalization yields an $L^2$ (possibly non-Gaussian) limit.

Therefore, the decay rate of the “error” in the first-moment asymptotics decides whether CLT-type (Gaussian) fluctuations arise (Yang, 23 Mar 2025, Miłoś, 2012, Ren et al., 2014).

In particular, for the supercritical Ornstein-Uhlenbeck superprocess (with generator $L = (1/2)\sigma^2 \Delta - \mu x\cdot\nabla$ and equation $u_t = L u + \alpha u - \beta u^2$), the fluctuation limit (after centering and normalization) shows a phase transition between Gaussian and non-Gaussian limits, determined by the relative sizes of the exponential growth (parameter $\alpha$) and the Ornstein-Uhlenbeck drift (parameter $\mu$) (Miłoś, 2012).

4. Backbone and Skeleton Decompositions, Martingale Techniques

A structural representation for supercritical non-local superprocesses arises from the backbone (skeleton) decomposition (Murillo-Salas et al., 2014, Fekete et al., 2019, Chen et al., 2017). The process $X_t$ may be decomposed pathwise into:

• a backbone (or skeleton) $Z_t$, a branching particle system encoding genealogies of prolific (non-extinct) individuals,
• and a “dressing” of immigrating superprocess excursions, which are copies of the original process conditioned to die out, grafted continually along the backbone's paths.

For non-local branching, the backbone branching mechanism includes both local (offspring at parent location) and non-local (offspring displaced via a transition kernel $w(x,dy)$) components. The immigration law is described using Dynkin–Kuznetsov $N$-measures, which encode the excursion law of the superprocess and the intensity/rates for grafting conditioned-to-die superprocesses along the backbone’s paths (Murillo-Salas et al., 2014, Fekete et al., 2019).

Skeleton decompositions underpin strong convergence proofs and LLNs, as the long-term behavior of the entire superprocess is essentially governed by the backbone process, which simplifies analysis by localizing the non-extinct mass (Chen et al., 2017).

5. Regime-Dependent Strong Limit and Fluctuation Laws

Precise normalization and fluctuation results in supercritical non-local superprocesses are determined by second moment and $L\log L$ criteria (Ren et al., 2017, Ren et al., 2018, Liu et al., 2021). If the $L\log L$ condition fails, classical martingales tend to zero, and proper norming functions $\gamma_t$ are required to recover non-trivial limits. The limit variable $W$ has strictly positive density and compound Poisson structure for a wide class of non-local branching mechanisms (Ren et al., 2018). Tail asymptotics for $W$, including small-value and large-value behavior, are established, revealing analogues of Schröder and Böttcher regimes from branching process theory in this non-local and measure-valued context.

Almost sure fluctuation rates for eigenfunction-martingales and linear functionals are given by laws of the iterated logarithm (LIL). For eigenpairs $(\lambda, g)$ with eigenfunction $g$ and eigenvalue $\lambda$, let $$W_t(\lambda,g) = e^{-\lambda t} \langle g, X_t \rangle$$. Then, under a fourth-moment condition, three regimes are identified (Hou et al., 18 Aug 2025):

• Re$(\lambda) < \lambda_1/2$: LIL holds with scaling $e^{(Re(\lambda)-\lambda_1/2)t}/\sqrt{\log t}$
• Re$(\lambda) = \lambda_1/2$: LIL with scaling $1/\sqrt{t\log\log t}$
• Re$(\lambda) > \lambda_1/2$: Martingale converges, and the LIL characterizes error decay between $W_t(\lambda,g)$ and its almost sure limit.

The explicit expressions for the LIL limits involve second-moment functionals computed against the eigenmeasure $\widetilde{\varphi}$, principal martingale limits $W_\infty^\varphi$, and spectral data. For test functions formed as combinations of eigenfunctions, the LIL identifies, via spectral decomposition, the respective contributions of “small,” “critical,” and “large” eigenvalues (Hou et al., 18 Aug 2025).

A table summarizing these scaling regimes appears below:

Regime	Scaling Factor	LIL Limit Expression
$Re(\lambda) < \lambda_1/2$	$e^{(Re(\lambda)-\lambda_1/2)t}/\sqrt{\log t}$	Explicit in terms of variances and $W_\infty^\varphi$
$Re(\lambda) = \lambda_1/2$	$1/\sqrt{t\log\log t}$	Similar, extra intensity for real $\lambda$
$Re(\lambda) > \lambda_1/2$	$e^{(Re(\lambda)-\lambda_1/2)t}/\sqrt{\log t}$ applied to limit error	As above

These LILs generalize known results for multitype branching processes [Asmussen 1977], and are the first of their kind for supercritical non-local superprocesses (Hou et al., 18 Aug 2025).

6. Applications, Examples, and Functional Models

The theoretical framework covers wide model classes:

• Supercritical Ornstein-Uhlenbeck superprocesses, where the generator is $L = (1/2)\sigma^2\Delta - \mu x\cdot\nabla$ and the equation is $u_t = L u + \alpha u - \beta u^2$ (Miłoś, 2012, Ren et al., 2019).
• Branching Gaussian systems, including those with long memory (fractional Brownian or Ornstein-Uhlenbeck processes), for which LLNs and limiting spatial profiles reflect both deterministic shifts (“mutation”) and spatial dispersion (Kouritzin et al., 2016).
• Multitype superdiffusions in bounded domains, with both spatial and type-structure, in which analogous Perron-Frobenius theory applies and all fluctuation and LIL results follow (Yang, 23 Mar 2025).

Additionally, PDE models coupling local and nonlocal operators in supercritical regimes arise from Dirichlet and Neumann problems with supercritical nonlinearities (Amundsen et al., 2023, Amundsen et al., 2023). In these models, variational methods on convex sets restore compactness, enabling proof of existence and characterization of spatial patterns even with nonlinearities above the critical Sobolev exponent, further linking PDE and probabilistic superprocess perspectives.

7. Unifying Themes and Outlook

The unified probabilistic and analytic framework for supercritical non-local superprocesses is built on the interplay of spectral theory, martingale convergence, and structural decompositions (backbone, skeleton, spine), which together uncover both average and fluctuation phenomena:

• Spectral gap and eigentriple structure govern all limit theorems and fluctuation regimes.
• Non-locality in branching leads to significant modifications in backbone components, immigration rates, and martingale functionals (Murillo-Salas et al., 2014, Fekete et al., 2019).
• Second and fourth moment assumptions determine the qualitative nature (Gaussian vs. non-Gaussian, polynomial vs. exponential scalings) of the limit and fluctuation laws.
• The development of functional limit theorems (FCLT, LIL) now mirrors in full generality the classical results for Galton–Watson and multitype branching processes.

This framework accommodates applications to spatial population models, ecological and epidemiological systems with rare but dominant “super-individuals” (Foucart et al., 2016), and stochastic PDEs in nonlocal and supercritical parameter regimes. The methodology is robust, extending from particle system approximations to continuous measure-valued models, and includes both essentially local and fully non-local spatial settings. The extension to infinite variance branching, criticality, and the structure of record processes opens further analytical and probabilistic directions for the paper of extreme events and spatial population clustering (Cardona-Tobón et al., 23 Jul 2025).