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

Rachid Belfadli, Youssef Ouknine, Ercan Sönmez

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.

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.
Paper Structure (7 sections, 3 theorems, 35 equations)

This paper contains 7 sections, 3 theorems, 35 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Fractional calculus
  4. Fractional Brownian sheet
  5. Girsanov theorems
  6. Existence of a weak solution
  7. Existence of the processes $u$ and $v$.

Key Result

Theorem 2.1

With the notations above, assume that $\int_{[0, \cdot]} u_{\zeta} d \zeta \in I^{\alpha + \frac{1}{2}, \beta + \frac{1}{2}}_{0^{+}} (L^2)$ almost surely. Set $\psi_z:=(\mathcal{K}^{\alpha, \beta})^{-1} \left( \int_{[0, \cdot]} u_{\zeta} d \zeta \right)(z)$ and consider the shifted process defined ). If satisfies $\mathbb{E}[L_T]=1$, then the process $\{\tilde{B}^{\alpha,\beta}_z, \, z\in [0, T

Theorems & Definitions (5)

  • Theorem 2.1
  • Theorem 2.2
  • Remark 2.3
  • Definition 3.1
  • Theorem 3.2