Unified Multifractal Framework

• Unified Multifractal Framework is a comprehensive approach that integrates cascade structures, nonlinear scaling laws, and tensor network constructions to characterize a wide range of multifractal phenomena.
• It leverages advanced methodologies like MERA networks and Legendre duality to generate, estimate, and simulate both canonical and non-Gaussian multifractal processes efficiently.
• The framework extends to discrete and functional settings, applying large-deviation principles and multifractal tube formulas to connect theoretical insights with practical computational schemes.

A unified multifractal framework provides a comprehensive description bridging the stochastic, geometric, and algorithmic aspects of multifractal phenomena in measures and processes. It systematizes the representation, estimation, and generation of multifractal structures—ranging from canonical processes such as fractional Brownian motion and log-infinitely divisible cascades, to fully discrete count data, functional field representations, renormalization-inspired tensor network samplers, and practical computational schemes. The framework is marked by an overview of cascade structures, scaling laws parameterized by nonlinear spectra, Legendre duality, and efficient algorithmic realizations.

1. Network-Based Construction of Multifractal Processes

A mathematically explicit construction of unified multifractal processes deploys a tensor network formalism built on conditional Gaussian kernels and recursive binary-tree contractions. Each increment $\Delta X_{t,\tau_l}$ at time %%%%1%%%% and scale $\tau_l$ is generated by merging two finer-scale increments with a transfer weight $m_{\tau_l}^{\tau_{l+1}}$ and added Gaussian noise: $$p(\Delta X_{t,\tau_l} \mid \Delta X_{t,\tau_{l+1}}=z_1, \Delta X_{t+\epsilon,\tau_{l+1}}=z_2) = \mathcal{N}\left(m_{\tau_l}^{\tau_{l+1}}(z_1+z_2), \sigma_{\tau_l}^2\right).$$ Organizing these via a binary tree ("light-cone"), each node at level $\tau_{l}$ is a Gaussian sum of its two children at $\tau_{l+1}$ plus independent noise. The process at the final ("IR") time scale is obtained by summing the leaves. In the deterministic limit, choosing

$$m_{\tau_l}^{\tau_{l+1}} = (\tau_{l+1}/\tau_l)^{H-1}$$

and appropriate $\sigma_{\tau_l}$, recovers the exact law and covariance of fractional Brownian motion (fBm). Non-Gaussian multifractality is introduced via randomization of $m_{\tau_l}^{\tau_{l+1}}$ using a multiplicative cascade $M(\tau)$ with the law-invariance property

$$M(a\,\tau) \stackrel{\text{law}}{=} M_1(a) M_2(\tau), \quad M_1(a)\perp M_2(\tau).$$

This framework thus encodes both classical fBm and its multifractal log-infinitely divisible generalizations within a single light-cone tensor network (Descamps, 2016).

2. Unified Scaling Laws, Legendre Duality, and Multifractal Spectra

The statistical essence of multifractality lies in nonlinear scaling exponents and their dual spectrum:

• For a measure $\mu$, the local Hölder exponent $\alpha(x)$ is defined via $\mu(B(x,r)) \sim r^{\alpha(x)}$.
• The moment partition function $Z(q,r) = \sum_i \mu_i(r)^q \sim r^{\tau(q)}$ yields the global scaling exponents.
• The multifractal spectrum $f(\alpha)$, giving the Hausdorff dimension of points with exponent $\alpha$, is provided by a Legendre transform: $$f(\alpha) = \inf_{q} (q\alpha - \tau(q)), \quad \alpha = \tau'(q)$$ (Salat et al., 2016, Abry et al., 2012).

The same formal structure underpins generalized, mixed, and higher-dimensional spectra—in dynamical systems via topological pressure and free energy, in mixed–$\varphi$ frameworks, and large-deviation-based formulations (Climenhaga, 2011, Makhmudov et al., 2024, Menceur et al., 2018).

3. Algorithmic Realizations: MERA Networks and Efficient Sampling

A significant advance in the unified framework is the deployment of the Multi-scale Entanglement Renormalization Ansatz (MERA), a deep tensor network originated in quantum lattice models, for efficient path sampling of multifractal processes:

• Construct a binary MERA network with depth $L \sim \log_2 N$, where $N$ is the number of output time steps.
• Calibrate isometries and disentanglers so that the two-point statistics of the sampled process match fBm or its multifractal extensions.
• Sampling proceeds by forward propagation of i.i.d. Gaussians at the apex through the layers, extracting increments at the leaves.
• The overall computational complexity is $O(N\log N)$, outperforming $O(N^2)$–$O(N^3)$ classical Cholesky or circulant methods (Descamps, 2016).

4. Extension to Discrete and Functional Settings

The unified multifractal framework encompasses not only real-valued, but also integer-valued processes. Through the thinning operator $\odot$, which parallels continuous scalar multiplication,

• Integer-valued multifractal processes are constructed as time-changed compound Poisson processes using nondecreasing multifractal clocks $Y(t)$ (subordinators with multifractal scaling).
• The scaling of moments matches the classical theory: factorial moments scale exactly as $t^{\tau(n)}$, with $\tau(n)$ inherited from the continuous multifractal clock (Grahovac, 26 Sep 2025).

In the functional (field-theoretic) context, a superposition of Gaussian subensembles, each labeled by a local Hölder exponent $h$, with weights $\mu(h)$ encoding the singularity spectrum $D(h)$, yields a characteristic functional

$$\Phi[\alpha] = \int dh\,\mu(h)\,\exp\left(-\frac{1}{2} \int \alpha(x) C_h(x-x') \alpha(x') dx\,dx'\right)$$

which recovers all moment and PDF statistics in a unifying, tractable form. The scaling exponents are then

$\zeta_q = \min_h [qh + 1 - D(h)]$

(Warnecke et al., 23 Sep 2025).

5. Generalizations: Dynamical, Nonconcave, and Mixed Spectra

Recent generalizations address broader classes of systems and estimation challenges:

• For dynamical multifractal fields, stochastic partial differential equations with linear (energy transport) and multiplicative chaos (intermittency) terms generate fields with prescribed multifractal spectra, including log-normal spectra matched to turbulence (Apolinário et al., 2021).
• Unified formalism accommodates nonconcave multifractal spectra via lifted Legendre transforms, enabling accurate estimation of complex spectra from empirical data using generalized wavelet-leader techniques (Leonarduzzi et al., 2018).
• In fully developed turbulence, joint (longitudinal and transverse) multifractal formalism is employed, leading to bivariate spectra $D(h_1, h_2)$ and explicit relations between inertial-range and dissipation-range statistics, supported by large-scale numerical simulations (Buaria, 18 Jan 2026).

6. Analytical and Large-Deviation Foundations

At the core of the unified framework is the equivalence between the multifractal formalism and large deviation theory. Under general conditions (exponential tightness, existence of free energy, uniqueness of equilibrium states), multifractal spectra are directly obtainable as Legendre transforms of large-deviation rate functions: $$f(\alpha) = \inf_{q} (\tau(q) - q\alpha) + h_{\text{top}}(T,\Omega)$$ General results extend from classical additive to nonadditive, mixed, and non-Gibbs settings, encompassing essentially all known multifractal spectra from dynamical systems and statistical physics (Makhmudov et al., 2024, Climenhaga, 2011, Menceur et al., 2018).

7. Geometric and Computational Extensions: Multifractal Tubes

The framework permits geometric quantification through multifractal tube formulas—generalizing Minkowski volumes to $q$-weighted measures, with explicit asymptotics and explicit formulas via renewal theory and multifractal zeta-functions. This yields multifractal analogues of Steiner formulas for self-similar measures, tightly linking multifractal dimensions to geometric content (Olsen, 2013).

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The unified multifractal framework thus provides an interconnected architecture for the representation, analysis, and simulation of multifractal objects, applicable across continuous, discrete, dynamical, and functional settings; it leverages renormalization-inspired tensor networks, large-deviation principles, advanced computational techniques, and a broad spectrum of theoretical and empirical methodologies (Descamps, 2016, Grahovac, 26 Sep 2025, Salat et al., 2016, Warnecke et al., 23 Sep 2025, Leonarduzzi et al., 2018, Apolinário et al., 2021, Climenhaga, 2011, Makhmudov et al., 2024, Menceur et al., 2018, Olsen, 2013).