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

TL;DR

This paper proves the existence and uniqueness of solutions for a stochastic PDE driven by two correlated fractional Brownian sheets, demonstrating how additive noise can regularize such equations under weak conditions.

Contribution

It introduces a novel approach combining two-parameter fractional calculus and Girsanov's theorem to handle correlated fractional noises in stochastic PDEs.

Findings

Existence and uniqueness of strong solutions established

Additive noise regularizes the stochastic PDE

Weak assumptions on the drift are sufficient for well-posedness

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.