Hierarchical symmetry selects log-Poisson cascades: classification, uniqueness, and stability

Abstract: Within i.i.d. multiplicative cascades, a single axiom -- the hierarchical symmetry, a linear contraction on incremental scaling exponents -- is shown to be necessary and sufficient for the cascade multiplier to be log-Poisson. We establish three results: (1) a characterization theorem proving that the hierarchical symmetry uniquely determines the log-Poisson distribution with explicit parameters; (2) a classification theorem proving that the hierarchical symmetry selects exactly the log-Poisson class from the full log-infinitely-divisible family, excluding log-normal, log-stable, and all intermediate generators; and (3) a stability theorem proving that approximate hierarchical symmetry implies approximate log-Poisson, with an explicit $O(\sqrt{\varepsilon})$ Wasserstein bound. The proofs reduce the problem to the Hausdorff moment problem on $[0,1]$ via the change of variables $u = e^{kx}$, where determinacy and stability follow from classical results.

• The paper rigorously demonstrates that imposing hierarchical symmetry uniquely selects log-Poisson laws in i.i.d. cascade models.
• It develops an explicit form for the scaling exponents and cascade multipliers, excluding all log-infinitely-divisible laws except log-Poisson.
• The stability theorem quantitatively shows that approximate symmetry leads to distributions that closely match the precise log-Poisson law.

Hierarchical Symmetry as a Rigorous Selector for Log-Poisson Cascades

Introduction

This work establishes a rigorous mathematical framework to address the selection mechanism for multiplicative cascade generators in the context of multifractal phenomena, most notably those encountered in turbulence and related complex systems. Building upon heuristic and physically motivated observations by prior studies, primarily those of She and Leveque, Dubrulle, and others, the paper delivers definitive classification, uniqueness, and stability results for the selection of log-Poisson distributions via a deterministic hierarchical symmetry axiom within the class of i.i.d. cascades.

Formal Framework and Main Axiom

The study considers i.i.d. multiplicative cascades, in which the observable scaling properties of a physical quantity across nested scales are governed by random multipliers $W$ with $\mathbb{E}[W] = 1$. The empirical statistics are encoded in scaling exponents $\zeta_p$ and corresponding incremental exponents $\delta_p = \zeta_{p+k} - \zeta_p$, derived from the structure functions $S_p(\ell) \sim \ell^{\zeta_p}$.

The central postulate (termed A1: Hierarchical Symmetry) asserts the existence of a contraction parameter $\beta \in (0,1)$ such that

$$\delta_{p+k} = (1 - \beta) \delta_\infty + \beta\,\delta_p$$

for all $p$ in a prescribed set. This linear recurrence encapsulates the empirical observation of consistent proportional scaling in the increments of the exponents and is the sole axiom considered for selection.

Characterization and Uniqueness of Log-Poisson Cascades

A remarkable result is achieved: under A1, the entire exponent sequence is forced into the form

$\zeta_p = \gamma p + C (1 - \beta^{p/k})$

with $\gamma, C$ fixed by observable scaling properties. The corresponding cascade multiplier is uniquely determined to be log-Poisson: $\mathbb{E}[W] = 1$0 where the parameters are explicit functions of observed structure function increments: $\mathbb{E}[W] = 1$1, $\mathbb{E}[W] = 1$2, $\mathbb{E}[W] = 1$3.

It is explicitly proven that no other log-infinitely-divisible law, including log-normal, log-stable, or mixtures thereof, can satisfy A1. The proof employs a reduction via the Hausdorff moment problem, leveraging determinacy, stability properties on compact sets, and coupling constructions to exclude possible alternative generators.

Classification in the Log-Infinitely-Divisible Family

Within the general log-infinitely-divisible family, the paper provides a rigorous dichotomy: the hierarchical symmetry axiom selects exactly those laws with L\'evy measure concentrated at a single negative value ($\mathbb{E}[W] = 1$4 with $\mathbb{E}[W] = 1$5), zero Gaussian component ($\mathbb{E}[W] = 1$6), i.e., the compound Poisson structure underlying log-Poisson. The proof excludes any generator with Gaussian component, positive jumps, or diffuse negative support.

The difference between typical turbulent cascades and classic log-normal models is made stark: the Carleman and Stieltjes moment problems are shown to diverge/converge exactly in the log-Poisson/log-normal cases, leading to moment (in)determinacy. Thus, exponent observations fully determine the cascade law in the log-Poisson case but not in the log-normal regime.

Stability: Robustness to Perturbations

A1's uniqueness result is complemented by a quantitative stability theorem: given any cascade whose scaling increments approximately satisfy the symmetry axiom (i.e., residuals $\mathbb{E}[W] = 1$7), the generating distribution is shown to lie at Wasserstein-1 distance $\mathbb{E}[W] = 1$8 from a precise log-Poisson law, with explicit, computable constants in terms of $\mathbb{E}[W] = 1$9. The proof employs compactness and Chebyshev bounds after mapping the L\'evy measure via $\zeta_p$0 to $\zeta_p$1, simplifying the moment stability analysis to an elementary calculation.

Implications and Corollaries

The derived multifractal spectrum has width $\zeta_p$2 and the symmetry parameter $\zeta_p$3 controls the entire multifractal curve shape under geometric or conservation constraints. Connections to applications are drawn, including the codimension of most singular structures and reduction in parameter dimension under conservation laws.

Critically, the established biconditional equivalence—A1 holds if and only if the multiplier is log-Poisson—eliminates ambiguity in structure function interpretations, bridging theoretical and empirical analyses for turbulence, rainfall, finance, and broader multi-scale systems.

Theoretical and Practical Scope

The framework sharply distinguishes the hierarchical symmetry as a powerful selector in multiphysics/complexity analysis. From a theoretical standpoint, it repositions the She-Leveque and Dubrulle symmetry heuristics as mathematically necessary and sufficient selection principles for cascade generator laws. This clarifies prior physical arguments and strengthens the universality assertion for multifractal settings.

The stability results enable robust data-driven parameter estimation, offering concrete tolerances for empirical testing of the log-Poisson hypothesis via observed scaling exponents, and supporting model inference even with mild real-world deviations from exact symmetry.

Directions for Future Investigation

The formalism is restricted to the i.i.d. case. Directions for further research include extension to stationary ergodic or Markovian multipliers (where L\'evy machinery may fail), analysis of extremal intermittent/monofractal boundary cases as $\zeta_p$4, and the development of adaptive, statistically robust procedures for hierarchy step $\zeta_p$5 selection and for practical quantification of the stability constants in experimental data analysis.

Conclusion

This paper demonstrates that hierarchical symmetry is both necessary and sufficient to select the log-Poisson class for multiplicative cascade generators within the i.i.d. paradigm, with strong uniqueness and robust stability properties. The results formalize, unify, and extend the theoretical underpinnings for a wide range of multifractal analyses and provide reliable machinery for practical inference of cascade laws from empirical scaling data.

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