Natural Multiplicative Chaotic Measure

• Natural multiplicative chaotic measure is defined as the scaling limit of exponentiated log-correlated fields with appropriate renormalization to yield multifractal structures.
• It exhibits distinct regimes—subcritical, critical, and supercritical—each characterized by unique properties such as non-atomic behavior and atomic decomposition under phase transitions.
• The measure demonstrates universality through invariant approximation schemes and harmonic analysis applications, providing insights into fractal geometry and statistical physics.

A natural multiplicative chaotic measure is a canonical random measure obtained by exponentiating an underlying log-correlated random field (possibly Gaussian or non-Gaussian), followed by suitable renormalization. Such measures arise at the interface of probability, harmonic analysis, statistical physics, and fractal geometry, providing models for random multifractal structures with deep universality properties. The terminology encompasses the full class of measures constructed by the multiplicative chaos paradigm, subject to natural regularity, moment, and invariance requirements, and is now rigorously understood both in the classical Gaussian regime and beyond to non-Gaussian, subcritical, critical, and supercritical phases.

1. Foundational Construction

A natural multiplicative chaotic measure $\mu$ is built as the scaling limit of regularized exponential functionals of a log-correlated field. For a bounded domain $D \subset \mathbb{R}^d$ and a centered Gaussian field $X$ with covariance

$\E[X(x)X(y)] = -\log|x-y| + g(x,y), \quad g \in H^s_{\text{loc}}(D\times D),\ s>d,$

one constructs a mollified approximation $X_\varepsilon = \phi_\varepsilon*\!X$ using a smooth compactly supported $\phi$ with vanishing moments up to $d-1$. The core regularized measure is then

$\mu_\varepsilon^\gamma(dx) = \exp\left[\gamma X_\varepsilon(x) - \frac{\gamma^2}{2} \E[X_\varepsilon(x)^2]\right]\, dx,$

and the (subcritical or critical) chaotic measure $\mu$ is defined as the limit, in law or probability, as $\varepsilon \to 0$ under appropriate normalization. In the supercritical phase $\gamma>\sqrt{2d}$, one further inserts singular scaling pre-factors to obtain a limiting atomic measure (Bertacco et al., 10 Apr 2025).

This construction generalizes to non-Gaussian log-correlated fields under moment control, to higher or lower dimensions, and to abstract settings such as random cascades, random matrix point fields, zeta function models, and occupation fields of stochastic processes (Lin et al., 6 May 2025, Jego, 2018, Gorodetsky et al., 13 Mar 2025, Aïdékon et al., 2021, Ostrovsky, 2016). The canonical normalization and proof of convergence are model-dependent.

2. Regimes: Subcritical, Critical, and Supercritical

The properties of natural multiplicative chaotic measures hinge on the intermittency parameter $\gamma$, and the dimension $d$:

• Subcritical ($0 < \gamma < \sqrt{2d}$): The measure $\mu^\gamma$ is non-degenerate, non-atomic, and singular with respect to Lebesgue measure. Its construction as the exponential of the regularized field is stable under mollification and under various approximation schemes; it is characterized uniquely by its covariance structure and the normalization (Junnila et al., 2015).
• Critical ($\gamma = \sqrt{2d}$): The naive exponentiation yields the zero measure. Two equivalent renormalizations exist: (i) the derivative martingale and (ii) Seneta–Heyde scaling. Both yield a nontrivial, non-atomic measure $\mu'$ with multifractal scaling and with only finite moments of order $q<1$ (Duplantier et al., 2012).
• Supercritical ($\gamma > \sqrt{2d}$): A further scaling is required. The limiting measure $\mu_\gamma$ becomes purely atomic, described as an integrated Poisson point process over the critical chaos measure. Specifically, one realizes $\mu_\gamma$ as

$$\mu_\gamma = a_{\gamma,\phi} \int_{D\times(0,\infty)} z\, \eta(dx,dz),$$

where conditionally on the critical measure $\mu_{\gamma_c}$, $\eta$ is a Poisson process with intensity $\mu_{\gamma_c}(dx)\, z^{-1-\gamma_c/\gamma}\, dz$ (Bertacco et al., 10 Apr 2025).

3. Universality and Uniqueness

A central property of natural multiplicative chaotic measures is their universality and approximation invariance. For Gaussian fields, the law of the measure depends only on the covariance structure and not on the details of the mollifier or approximation procedures. In particular, for any two mollification schemes satisfying regularity and moment conditions, the limiting measure is unique up to a model-dependent constant factor (Junnila et al., 2015, Bertacco et al., 10 Apr 2025). In the non-Gaussian subcritical regime, an analogous universality holds in the sense of absolute continuity with Gaussian chaos measures, and with a strongly robust theory of couplings (Kim et al., 25 Oct 2024).

Furthermore, the martingale structure is pivotal in establishing both existence and uniqueness. When the field admits an associated positive martingale sequence of random measures, the limiting law is canonical and independent of auxiliary regularizations. In non-Gaussian settings, uniqueness is established via weak convergence in $L^r$ and modified second-moment methods (Gorodetsky et al., 13 Mar 2025).

4. Harmonic Analysis and Fourier Dimension

Natural multiplicative chaotic measures exhibit specific regularity in the harmonic-analytic sense, quantified by Fourier decay and Fourier dimension. Recent advances have unified the proof of sharp polynomial decay for a large class of such measures, including classical subcritical Gaussian multiplicative chaos, Mandelbrot random coverings, Poisson chaos, and multiplicative cascades (Lin et al., 6 May 2025). Under explicit $L^p$ and Hölder moment assumptions on the underlying multiplicative sequence, one obtains almost sure decay

$$|\widehat\mu(\xi)| = O(|\xi|^{-T}), \qquad \forall T < \dim_F \mu,$$

with the exact Fourier dimension computable by an explicit formula in terms of the process regularity and correlation exponents. For subcritical 1D/2D Gaussian chaos, the formula yields

$$\dim_F\mu_\gamma = d - \frac{\gamma^2}{2}, \qquad 0<\gamma<\tfrac{\sqrt{2d}}{2},$$

and matches the correlation/Hausdorff dimension, establishing the universality of the spectral and multifractal properties in this context (Lin et al., 6 May 2025).

5. Non-Gaussian and Arithmetic Chaos

Extensions of the theory accommodate non-Gaussian, but log-correlated, fields. Examples include multiplicative chaos for Steinhaus random multiplicative functions, random zeta functions, and occupation measures of Brownian motion. In these settings, chaos measures are shown to exist (at least in the $L^1$-regime or subcritical phase) using modified second-moment and Girsanov methods, with limit measures determined up to absolute continuity by the log-correlated structure (Gorodetsky et al., 13 Mar 2025, Saksman et al., 2016).

In arithmetic settings, the construction relates to the paper of Dirichlet series, random Euler products, and the statistical mechanics of prime-based random fields. These chaos measures retain multifractality, exhibit stable CLT behavior with random variance given by the total mass of chaos, and are independent of approximation schemes within suitable classes (Gorodetsky et al., 13 Mar 2025).

6. Multifractality, Extreme Sets, and Applications

Natural multiplicative chaotic measures are carried by exceptional or "thick" sets of the underlying field, i.e., points where the field exhibits extreme local behavior. For subcritical chaos, the measure is singular and supported on sets of precise Hausdorff dimension determined by the theory. The thick-point sets, their measure, and scaling limits serve both as constructive skeletons (through discrete approximations) and as tools for applying the chaos paradigm to discrete systems, such as random walks, loop soups, and random matrix theory. The connection to the geometry of extremes, KPZ-type relations, and large deviations is central for both theoretical and applied directions (Junnila et al., 2022, Jego, 2018, Jego, 2019).

In harmonic and spectral analysis, the properties of multiplicative chaos underpin the understanding of singular measures with polynomial Fourier decay, Rajchman property, and sets of multiplicity, especially in the context of classical covering problems (Tan, 12 Nov 2025).

7. Atomic, Non-Atomic, and Rajchman Regimes

Depending on regime, the natural multiplicative chaotic measure may be atomic (supercritical), non-atomic (subcritical/critical), or exhibit further analytic regularity. In the atomic case, the measure is determined via a Poisson random measure with intensity governed by the critical chaos. For non-atomic measures, Rajchman properties and Fourier decay (vanishing Fourier coefficients at infinity) are generic, ensuring singularity and multifractal structure; for cascade constructions, the non-atomicity is preserved provided standard uniform integrability and moment bounds are satisfied (Bertacco et al., 10 Apr 2025, Tan, 12 Nov 2025).

The presence or absence of atoms reflects the interplay between multiplicative intermittency and phase transitions—central to the allocation of mass and the decomposition via Poissonizers in the supercritical regime.

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References

• "Uniqueness of supercritical Gaussian multiplicative chaos" (Bertacco et al., 10 Apr 2025)
• "The non-covered set in Dvoretzky covering is a set of multiplicity" (Tan, 12 Nov 2025)
• "Harmonic analysis of multiplicative chaos Part II: a unified approach to Fourier dimensions" (Lin et al., 6 May 2025)
• "Multiplicative chaos measure for multiplicative functions: the $L^1$-regime" (Gorodetsky et al., 13 Mar 2025)
• "Absolute continuity of non-Gaussian and Gaussian multiplicative chaos measures" (Kim et al., 25 Oct 2024)
• "The uniqueness of the Gaussian multiplicative chaos revisited" (Junnila et al., 2015)
• "Multiplicative chaos measures from thick points of log-correlated fields" (Junnila et al., 2022)
• "Planar Brownian motion and Gaussian multiplicative chaos" (Jego, 2018)
• "Characterisation of planar Brownian multiplicative chaos" (Jego, 2019)
• "Multiplicative chaos of the Brownian loop soup" (Aïdékon et al., 2021)
• "A Theory of Intermittency Differentiation of 1D Infinitely Divisible Multiplicative Chaos Measures" (Ostrovsky, 2016)