Fractional Brownian Sheet Overview

• Fractional Brownian Sheet is a multi-parameter Gaussian field defined by anisotropic Hurst parameters and a product-form covariance structure.
• It exhibits operator-self-similarity and stationary rectangular increments, making it a canonical model for anisotropic long-range dependence.
• Its applications span advanced stochastic analysis, simulation methods, and functional limit theorems in high-dimensional random field modeling.

A fractional Brownian sheet (fBs) is a continuous, centered Gaussian random field indexed by multi-dimensional time, characterized by anisotropic Hurst parameters. It generalizes the classical fractional Brownian motion to higher dimensions and exhibits a rich combination of operator-self-similarity, stationary rectangular increments, and covariance structure that factorizes along coordinates. Fractional Brownian sheets serve as canonical models for anisotropic long-range dependence, are pivotal for functional limit theorems, and underpin diverse applications in stochastic analysis, statistical estimation, and numerical simulation (Makogin et al., 2014, Wang, 2011, Wu et al., 2015).

1. Mathematical Definition and Key Properties

A $d$-parameter fractional Brownian sheet $B_H = \{ B_H(t) : t \in [0,1]^d \}$ is determined by a Hurst vector $H = (H_1, \dots, H_d) \in (0,1)^d$ and has covariance

$$\mathbb{E}[B_H(s) B_H(t)] = \prod_{i=1}^d \frac{1}{2} \left( |s_i|^{2H_i} + |t_i|^{2H_i} - |t_i - s_i|^{2H_i} \right).$$

Increments along arbitrary rectangles are stationary: $$\Delta_h B_H(t) = \sum_{\epsilon \in \{0,1\}^d} (-1)^{d - |\epsilon|} B_H(t + \epsilon \cdot h)$$ with laws of increments depending only on $h$, not the location. Operator-self-similarity holds: $$\{ B_H(c \cdot t) : t \in [0,1]^d \} \stackrel{d}{=} \{ \left( \prod_{i=1}^d c_i^{H_i} \right) B_H(t) : t \in [0,1]^d \}.$$ The sheet is uniquely determined as a Gaussian field by its product-form covariance (Makogin et al., 2014, Wang, 2011, Wu et al., 2015).

2. Integral Representations and Simulation

Analytically, $B_H$ admits a Wiener-type double integral representation. In two parameters,

$$B_H(s, t) = \iint_{[0,s] \times [0,t]} K_{H_1}(s, u) K_{H_2}(t, v) \, dW_{u,v}$$

where $W$ is a standard Brownian sheet, and $K_{H_i}$ are the Volterra fractional kernels analogous to Mandelbrot–Van Ness (Bardina et al., 2020).

Approximate simulation is realized by discretizing via Lévy sheets or in the spectral domain. For instance, Bardina–Rovira construct sequences $X^n$ of Wiener-type integrals against Lévy-sheet-generated kernels converging weakly (in law) to the fBs in $C([0,1]^2)$ (Bardina et al., 2020). Weighted tensorized and operator scaling variants yield more general textures, interpolating between pure fBs and isotropic fractional fields (Esser et al., 2024).

3. Functional Limit Theorems and Discrete Models

Fractional Brownian sheets arise as scaling limits for sums of dependent random fields and for random partition-based discrete analogues. For example, central limit and invariance principles establish weak convergence in $D([0,1]^d)$ of suitably normalized partial sums to fBs, under product-form coefficients and physical dependence conditions (Wang, 2011).

Durieu–Wang introduce three discrete random partition models on $\mathbb{N}^2$ whose partial sums converge to fBs with full range of Hurst indices, providing a discrete counterpart to the fBs comparable to the relation between random walk and Brownian motion (Durieu et al., 2017). Such functional limit theorems underpin modern statistical and probabilistic analysis of high-dimensional random fields.

4. Covariance Structure, Self-Similarity, and Extensions

The covariance factorization is the key structural feature distinguishing the fBs from general self-similar, rectangular-increment stationary Gaussian fields. Notably, Makogin–Mishura show that self-similarity and rectangular-increment stationarity alone do not uniquely specify the product-form covariance: it is possible to construct other fields sharing these two properties but whose covariance deviates from the fBs form unless additional constraints are imposed (cf. Lamperti transform and its product-form stationarity) (Makogin et al., 2014).

Extensions include multifractional sheets and operator fractional Brownian sheets of Riemann-Liouville type; here, the scaling law applies to operator-valued exponents, yielding more complex anisotropy and causality, as well as martingale-difference approximation frameworks (Dai et al., 2015, Ralchenko et al., 2011).

5. Regularity, Geometric Properties, and Image Sets

Sample paths of fBs exhibit anisotropic Hölder continuity determined by the Hurst parameters. Regularity results show that, for $d \ge 2$ and $\bar{H} = (H_1 + \cdots + H_d) / d > (d-1)/d$, almost all sample paths are strongly chargeable, i.e., they are primitives of divergences of continuous vector fields (Bouafia et al., 2024). This is tightly linked to the ability to define stochastic pathwise integration in higher dimensions.

With respect to fractal geometry and image sets, the Hausdorff dimension of $B^H(F+t)$ for $(N,d)$-fBs satisfies a uniform formula in the high-dimensional regime ($d \ge \sum_\ell 1/H_\ell$), and a weak uniform dimensional result for almost all translations $t$ when $d < \sum_\ell 1/H_\ell$. Refined sectorial and anisotropic local nondeterminism bounds are instrumental in proving these results (Balança, 2015).

6. Statistical Estimation, Variations, and Hermite Regimes

Statistical estimation of the Hurst vector is achieved by multi-dimensional wavelet analysis coupled with least squares regression. These estimators are asymptotically normal, efficient for $H_i \geq 1/2$, and exhibit modest bias for $H_i < 1/2$ (Wu et al., 2015).

Functional limit theorems for generalized variations (including power and Hermite variations) distinguish between Gaussian and non-Gaussian (Hermite sheet) regimes, depending on the Hermite rank of the variation functional and the minimal Hurst parameter component. In the non-Gaussian regime, limits are described by multiple Wiener–Itô integrals yielding Hermite sheets with self-similar and rectangular-increment stationary properties (Pakkanen et al., 2014, Reveillac et al., 2010).

7. Applications, SDEs, and Simulation

Fractional Brownian sheets are central in stochastic partial differential equations, particularly as input noise for SPDEs and for studying rough path and stochastic calculus in the plane. Well-posedness of SDEs driven by fBs with integrable drift has been proven under minimal Hurst parameter constraints exploiting sectorial local nondeterminism and compactness in Malliavin calculus (Bogso et al., 2023).

Discrete, continuous, and operator-scaling simulations leverage the spectral representation, Lévy sheet approximations, and martingale-difference constructions to realize fBs in practice with prescribed anisotropic regularity and long-range dependence. These methodologies support Monte Carlo schemes and statistical inference in high-dimensional stochastic models (Esser et al., 2024, Bardina et al., 2020).

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Table: Essential Properties of the Fractional Brownian Sheet

Property	Mathematical Statement	Reference
Covariance	$$\prod_{i=1}^d \tfrac{1}{2}(s_i^{2H_i} + t_i^{2H_i} - |s_i - t_i|^{2H_i})$$	(Makogin et al., 2014)
Self-similarity	$$B_H(c \cdot t) \overset{d}{=} \left( \prod c_i^{H_i} \right) B_H(t)$$	(Wang, 2011)
Stationary rectangular increments	Law of increments depends only on $h$, not on $t$	(Makogin et al., 2014)
Pathwise regularity	Hölder-continuity in each coordinate $< H_i$	(Bouafia et al., 2024)
Invariance principle	Weak convergence of partial sums to fBs in $D([0,1]^d)$	(Wang, 2011)

The fractional Brownian sheet is a fundamental object in multiparameter Gaussian processes, stochastic analysis, and mathematical modeling of spatial-temporal dependencies, characterized by its unique product-form covariance, rich regularity, and versatility in simulation and applications.