Two-dimensional fractional Brownian motion: Analysis in time and frequency domains

TL;DR

The paper addresses modeling multidimensional anomalous diffusion with interdependent components by introducing a two-dimensional fractional Brownian motion with a matrix-valued Hurst operator. It builds two formulations, a causal and a well-balanced version, using correlated Gaussian noises to realize direction-dependent scaling and cross-dependence, and derives complete time-domain and frequency-domain characterizations, including auto- and cross-covariances and PSDs for both the process and its increments. Explicit cross-covariance formulas, cross-PSDs, and their asymptotics are provided, with numerical simulations validating the theory and illustrating how the parameters $H_1$, $H_2$, and $\rho$ shape diffusion and spectral content. The causal model exhibits time asymmetry and a complex cross-PSD, while the well-balanced model is time-reversible with a real cross-PSD; together these constructions offer a flexible framework for multidimensional self-similar processes in fields such as biology, finance, and physics, with potential generalization to higher dimensions.

Abstract

This article introduces a novel construction of the two-dimensional fractional Brownian motion (2D fBm) with dependent components. Unlike similar models discussed in the literature, our approach uniquely accommodates the full range of model parameters and explicitly incorporates cross-dependencies and anisotropic scaling through a matrix-valued Hurst operator. We thoroughly analyze the theoretical properties of the proposed causal and well-balanced 2D fBm versions, deriving their auto- and cross-covariance structures in both time and frequency domains. In particular, we present the power spectral density of these processes and their increments. Our analytical findings are validated with numerical simulations. This work provides a comprehensive framework for modeling anomalous diffusion phenomena in multidimensional systems where component interdependencies are crucial.

Figures (7)

• Figure 1: The dependence between correlation $\rho$ of the underlying noise and (a) cross-correlation coefficient $\rho_{12}$ and (b) asymmetry parameter $\eta_{12}$ of the process $\mathbf{X}(t)$ and $\mathbf{X}^*(t)$. Different solid lines correspond to different Hurst exponents of the coordinates, while the black dashed line corresponds to the identity cases $\rho_{12}=\rho$ or $\eta_{12}=\rho$, shown as a guide to the eye.
• Figure 2: Sample trajectories of causal 2D fBm with $H_1=H_2=H$. Each panel corresponds to a given set of parameters $(H, \rho_{12})$, where the top row (a-c) corresponds to uncorrelated components and the bottom row (d-f) to cross-correlation $\rho_{12}=0.5$. Pairs of panels in each column have the same $H$. The trajectories are shifted on the horizontal axis for clarity.
• Figure 3: Sample trajectories of causal 2D fBm with different Hurst exponents, $H_1 \neq H_2$. Each panel corresponds to a given set of parameters $(H_1, H_2, \rho_{12})$, where the top row (a-c) corresponds to uncorrelated components and the bottom row (d-f) to cross-correlation $\rho_{12}=0.5$. The trajectories are shifted on the horizontal axis for clarity.
• Figure 4: Cross-covariance function depending on $H_1, H_2$ and $\rho_{12}=0.5$ for $H_1=H_2$. (a) $H_1 = H_2 = 0.2$, (b) $H_1 =H_2=0.5$, and (c) $H_1 = H_2 = 0.7$. The solid lines represent the causal version of the model and dashed lines the well-balanced case. Markers correspond to the estimated values of the cross-covariance function; circles correspond to the causal case, squares to the well-balanced case. Here, casual and well-balanced cases overlap, as for $H_1=H_2$ the model is time-reversible regardless of the definition (cf. Eq. (\ref{['eq:eta12']})). [id=MB]Increments are taken over a unit time interval, i.e., $\Delta \equiv \Delta^{\delta=1}$.
• Figure 5: Cross-covariance function depending on $H_1, H_2$ and $\rho_{12}$ for $H_1\neq H_2$. (a) $H_1 = 0.2, H_2=0.5$, (b) $H_1 = 0.2, H_2=0.7$, and (c) $H_1 = 0.5, H_2 = 0.7$. Solid lines represent the causal version of the model, the dashed lines the well-balanced case. Markers correspond to the estimated values of the cross-covariance function; circles correspond to the causal case, squares to the well-balanced case.

Theorems & Definitions (29)

• Definition 1: Causal 2D fBm
• Remark 1
• Definition 2: Well-balanced 2D fBm
• Theorem 1
• proof
• Remark 2
• Remark 3
• Remark 4
• Remark 5
• Theorem 2