Regularization of Hyperbolic Stochastic Partial Differential Equations By Two Fractional Brownian Sheets

Abstract: In this paper, we establish existence and uniqueness of strong solutions for a stochastic differential equation driven by an additive noise given by the sum of two correlated fractional Brownian sheets with different Hurst parameters. Our analysis relies on techniques from two-parameter fractional calculus and a tailored version of Girsanov's theorem. The main challenge arises from the correlation between the two noises and the technical requirements for applying Girsanov's theorem in this setting. We show that, despite these difficulties, the additive noise regularizes the equation, allowing well-posedness under weak assumptions on the drift.