DOI: 10.11648/j.sjams.20261404.12 ISSN: 2376-9513

Stochastic Integration with Respect to Fractional Brownian Sheets and Applications to Anisotropic SPDEs

Bou Diop
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This paper develops a comprehensive framework for stochastic integration with respect to multidimensional fractional Brownian sheets, with particular emphasis on the anisotropic setting where each Hurst index exceeds one-half. We introduce a regularizationbased approach that accounts for the directional structure of the sheet and establish its equivalence to the Malliavin calculus construction. Building on this foundation, we obtain wellposedness and regularity results for a class of stochastic partial differential equations driven by such sheets. The regularization method provides an intuitive interpretation of the integral while preserving the directional heterogeneity inherent in anisotropic models. We prove that the regularized integral converges in the mean-square sense to the Skorokhod integral, establishing a duality relation that connects our constructive approach to established analytical frameworks. Our well-posedness results are obtained through a fixed-point argument in suitably chosen solution spaces, while the regularity analysis relies on anisotropic versions of the classical Kolmogorov criterion. Notably, the H¨older exponents we obtain directly reflect the directional smoothness of the driving sheet. This feature has practical implications for numerical discretization and statistical estimation. Our findings extend known one-dimensional theories to genuinely multidimensional and anisotropic regimes, offering new tools for modelling systems with long-range dependence and directional heterogeneity. Applications include fluid flows with directional turbulence, financial markets with correlated assets exhibiting different memory properties, and biological tissues with anisotropic diffusion characteristics.

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