Supercritical Non-Local Branching Markov Process

Updated 22 August 2025

Supercritical non-local branching Markov process is a stochastic system where particles move spatially and reproduce with offspring potentially relocating far from their parent, leading to exponential growth.
The framework utilizes spectral theory and martingale techniques to establish laws of large numbers, central limit theorems, and iterated logarithm laws, providing detailed insights into long-term behavior.
Applications include ecological population modeling, genetic branching, and random graph analysis, supported by advanced analytic tools like spine and backbone decompositions.

A supercritical non-local branching Markov process is a stochastic system in which individual entities (particles or mass elements) undergo spatial motion and reproduce with location-dependent, potentially non-local mechanisms, such that the expected total mass or number of particles grows exponentially. The “non-local” aspect means that offspring may be placed at spatial positions other than that of the parent, and “supercritical” indicates that the leading eigenvalue of the process’s mean semigroup is strictly positive. Recent advances have developed a unified analytic and probabilistic framework for studying such processes, including rigorous laws of large numbers, central limit theorems, functional limit theorems, and laws of the iterated logarithm (LIL), for both particle and measure-valued settings.

1. Structural Framework and Definition

Let $X = (X_t)_{t\geq 0}$ denote a Markov branching process (possibly measure-valued; i.e., a superprocess) on a Luzin space $E$ . The system combines a base spatial motion $(P_t)$ —often a Feller process or diffusion—and a non-local branching mechanism, which is typically written (for test functions $f$ ) as

$\Psi(x,f) = -a(x)f(x) + b(x)f(x)^2 + \int_{\mathcal{M}_0(E)} \left(e^{-\langle f, \nu\rangle} - 1 + \langle f, \nu\rangle\right) H(x,d\nu),$

where $a(x), b(x)$ are measurable coefficients and $H(x,d\nu)$ is a kernel allowing for the possibility that offspring are born at locations distributed according to $\nu$ , possibly far from $x$ . The process is “supercritical” if the mean semigroup $T_t$ (governing $\mathbb{E}_\mu[\langle f, X_t\rangle] = \langle T_t f, \mu\rangle$ ) possesses a Perron–Frobenius triple: a principal eigenvalue $\lambda_1>0$ , strictly positive right eigenfunction $\varphi$ , and left eigenmeasure $\tilde{\varphi}$ satisfying $T_t\varphi = e^{\lambda_1 t}\varphi$ and $\langle T_t f, \tilde{\varphi}\rangle = e^{\lambda_1 t} \langle f, \tilde{\varphi}\rangle$ for all bounded measurable $f$ (Hou et al., 18 Aug 2025).

The non-locality of the branching mechanism is crucial: after reproduction, child particles (“mass”) may be instantaneously transported to positions unrelated to the parent, with distribution encoded in $H(x,d\nu)$ or related (for particle systems) terms in the offspring kernel (Li, 2012, Palau et al., 2018). This distinguishes such systems from classical (local) branching processes.

2. Laws of Large Numbers and Long-Time Growth

The fundamental result for supercritical systems is the law of large numbers (LLN), describing the typical exponential growth and asymptotic spatial profile. For a broad class of non-local (possibly measure-valued) branching Markov processes, and for any bounded measurable $f$ on $E$ , it holds that

$e^{-\lambda_1 t} \langle f, X_t\rangle \to \langle f, \tilde{\varphi}\rangle W_{\infty}$

in $L^2(P_\mu)$ and, under mild additional hypotheses, almost surely (Yang, 23 Mar 2025, Palau et al., 2018). Here $W_{\infty}$ is the almost-sure limit of the nonnegative martingale $W_t = e^{-\lambda_1 t} \langle \varphi, X_t\rangle$ , encapsulating the long-term random fluctuations in total mass, while $\tilde{\varphi}$ describes the deterministic spatial profile. The result is robust and extends to non-local Markov branching processes with spatially inhomogeneous motion—even with complex genealogical structures and in both particle and superprocess regimes (Li, 2012, Kouritzin et al., 2016). In multitype settings, the asymptotic proportion of each type is given by the corresponding coordinate of the principal left eigenvector (Kyprianou et al., 2017).

When the spatial motion is not Markov or the environment is random, appropriately modified LLNs still hold: for instance, in supercritical branching Volterra processes (with memory), the spatial distribution converges (after normalization) to a (random) Gaussian measure determined by the long-term characteristics of each particle’s trajectory (Kouritzin et al., 2016).

3. Fluctuations and Central Limit Theorems

Beyond almost-sure exponential growth, supercritical non-local branching Markov processes exhibit non-trivial fluctuations. The spectral gap between the principal eigenvalue $\lambda_1$ and the next-largest real parts of the spectrum of $T_t$ governs the precise limiting regime, leading to a trichotomy (Dean et al., 26 Feb 2025, Yang, 23 Mar 2025, Ren et al., 2013, Ren et al., 2014):

Gaussian Regime: If for a test function $f$ , a defined spectral decay rate $\epsilon(f) > \lambda_1/2$ , then the properly centered and normalized quantity

$e^{-\lambda_1 t} \langle f, X_t\rangle - \langle f, \tilde{\varphi}\rangle W_\infty$

(with normalization $e^{\lambda_1 t/2}$ ) converges in distribution to a non-degenerate Gaussian random variable, whose variance depends on the “second moment” coefficients of the branching mechanism and the projection of $f$ onto sub-leading eigenspaces (Yang, 23 Mar 2025, Ren et al., 2013, Ren et al., 2014).

Critical Regime: If $\epsilon(f) = \lambda_1/2$ , a further polynomial correction ( $t^{1/2}$ normalization) is needed, with the centered and normalized fluctuation converging to a Gaussian whose variance reflects the critical balance of modes.
$L^2$ or “Dominant Modes” Regime: If $\epsilon(f) < \lambda_1/2$ , the fluctuation, after normalization, converges in $L^2$ to a non-random limit dominated by contributions from slowly decaying spectral modes.

Functional central limit theorems extend these statements to the process level, establishing weak convergence of fluctuations (in distribution in $D[0,\infty)$ ) to Gaussian processes, with covariance structures determined by detailed spectral decompositions and the non-local component (Dean et al., 26 Feb 2025).

Non-Gaussian fluctuations (e.g., stable laws) arise in supercritical systems with heavy-tailed offspring distributions—the associated CLT is then replaced by convergence to $(1+\beta)$ -stable limits, with the exact normalization and qualitative behavior determined by the interplay of branching intensity, drift (from the spatial motion), and offspring tail index $\beta$ (Marks et al., 2018).

4. Laws of the Iterated Logarithm

The almost-sure fluctuation of linear functionals in these systems is captured by law of the iterated logarithm (LIL) type theorems (Hou et al., 18 Aug 2025, Hou et al., 19 May 2025). For $f$ a finite linear combination of eigenfunctions (possibly complex-valued) of $T_t$ , the real part of the associated martingale $W_t(\lambda,g)=e^{-\lambda t}\langle g, X_t\rangle$ satisfies:

If $\mathrm{Re}(\lambda) < \lambda_1/2$ , then

$\limsup_{t\to\infty} \frac{e^{(\mathrm{Re}(\lambda)-\lambda_1/2)t} \mathrm{Re}\, W_t(\lambda,g)}{\sqrt{\log t}} = C\sqrt{W_\infty^{(\varphi)}}$

almost surely for suitable $C > 0$ .

If $\mathrm{Re}(\lambda) = \lambda_1/2$ , the correct normalization is $\sqrt{t\log \log t}$ .
For linear functionals comprising several eigencomponents, after removing contributions from “large” eigenmodes ( $\mathrm{Re}(\gamma_k) > \lambda_1/2$ ), the remaining fluctuation after $e^{-\lambda_1 t/2}$ normalization has a LIL scaled by $\sqrt{2\log t}$ (or $\sqrt{t\log\log t}$ in the critical regime) (Hou et al., 18 Aug 2025, Hou et al., 19 May 2025).

These LIL regimes are robust under non-local branching, non-symmetric spatial motion, and can be formulated for both particle and superprocess cases.

5. Martingale Spines, Backbone Decomposition, and Genealogy

Pathwise genealogical structure and finer fluctuation results are elucidated via spine and backbone decompositions (Murillo-Salas et al., 2014, Jonckheere et al., 2017, Harris et al., 2022). The “spine” approach reweights the process via martingale changes of measure (e.g., Doob’s $h$ -transform), singling out a distinguished “immortal” lineage whose augmentation with independent immigrating subcritical processes reconstructs the full supercritical process in law. This backbone is intertwined with non-local branching events and conditionally Poissonian immigrations, providing both an explicit description of the process’s dynamics and a basis for representation formulas (many-to-one, many-to-few) for multi-time and multi-particle functionals. For non-local branching, these decompositions require additional ingredients to track the spatial displacements of offspring and are fundamental for establishing LLN, CLT, and LIL results, as well as for probabilistic constructions connecting to random geometry and genealogy (Murillo-Salas et al., 2014, Harris et al., 2022).

6. Spectral Theory and the Role of the Mean Semigroup

A pervasive theme is the essential role of the spectral analysis of the mean semigroup $T_t$ , whose leading eigenvalue $\lambda_1$ sets the exponential growth scale, and whose spectrum (eigenvalues and generalized eigenspaces) dictates the qualitative nature of fluctuations (Dean et al., 26 Feb 2025, Yang, 23 Mar 2025, Hou et al., 18 Aug 2025). Non-simple spectra (Jordan block phenomena, nilpotent contributions), non-self-adjointness (non-symmetric motion), and the complexification of eigenpairs (leading to oscillatory martingale limits) must be considered in the non-local, high-complexity regime (Ren et al., 2014, Hou et al., 18 Aug 2025).

The projections of test functions onto the eigenstructure determine the scaling and normalization of both LLN and all higher-order fluctuation theorems. For models with heavy tails or inhomogeneities, spectral gaps or tail exponents may govern the transition between Gaussian, stable, and deterministic limit regimes (Marks et al., 2018).

7. Applications, Generalizations, and Open Problems

Supercritical non-local branching Markov processes underpin a diverse range of applications: modeling population growth and dispersal (ecological and genetic branching), analysis of rare events in random graphs and surfaces (e.g., in random planar maps and Liouville quantum gravity (Gwynne et al., 9 Nov 2024)), spatially extended epidemics, and particle systems in disordered media. In random environments and with thresholds, branching parameters may themselves be driven by stochastic or deterministic exogenous dynamics, introducing further decoupling and limit theorem possibilities (Wu et al., 2023, Francisci et al., 2022).

Current research continues to generalize these results to processes with: non-Markovian underlying motion (e.g., branching fractional Brownian motion), infinite-variance offspring laws (yielding stable limit laws), lack of invariant measure (“ $\lambda$ -transient” settings), more irregular or discontinuous spatial motion, and to investigate large deviations, genealogy beyond limit theorems, and universality/fine structure at phase transitions between supercriticality and other criticalities (Kouritzin et al., 2016, Jonckheere et al., 2017).

Table: Key Regimes for Fluctuations

Regime	Spectral Relation	Normalization	Fluctuation Type
Gaussian	$\epsilon(f) > \lambda_1/2$	$e^{\lambda_1 t/2}$	Central Limit Theorem
Critical	$\epsilon(f) = \lambda_1/2$	$t^{1/2} e^{\lambda_1 t/2}$	CLT w/ polynomial
$L^2$ /Deterministic	$\epsilon(f) < \lambda_1/2$	Slower polynomial	$L^2$ convergence
Stable (heavy-tails)	Infinite variance branching	Model dependent	Stable law
LIL (small branching)	$\mathrm{Re}(\lambda) < \lambda_1/2$	$e^{(\mathrm{Re}\lambda-\lambda_1/2)t}\sqrt{\log t}$	Iterated logarithm
LIL (critical)	$\mathrm{Re}(\lambda) = \lambda_1/2$	$\sqrt{t \log\log t}$	Iterated logarithm
LIL (supercritical)	$\mathrm{Re}(\lambda) > \lambda_1/2$	Tail decay of martingale	Iterated logarithm

The theory thus offers a comprehensive suite of probabilistic and analytic tools—coupling spectral, martingale, and pathwise arguments—to analyze and characterize the rich world of supercritical non-local branching Markov processes and their measure-valued extensions.