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Stochastic Burgers equation driven by multiplicative Rosenblatt noise: local existence, uniqueness and regularity

Atef Lechiheb

Abstract

We study the stochastic Burgers equation driven by a multiplicative Rosenblatt noise with Hurst parameter $H \in (1/2,1)$. Using a fixed-point argument in a Malliavin--Sobolev space that controls the solution and its first two Malliavin derivatives, we prove local existence and uniqueness of a mild solution. We establish uniform moment bounds of all orders and prove Hölder regularity: spatial Hölder exponent $γ< 1/2$ and temporal Hölder exponent $α< H-1/2$, which are shown to be sharp by a lower bound for the linearized equation. The proof relies on sharp estimates of the heat kernel in the reproducing kernel Hilbert space $\cH$ of the Rosenblatt process, on Meyer's inequalities for moment bounds, and on a careful analysis of the Skorohod integral with respect to the Rosenblatt process. These results provide a rigorous foundation for the study of nonlinear SPDEs driven by non-Gaussian long-memory noise.

Stochastic Burgers equation driven by multiplicative Rosenblatt noise: local existence, uniqueness and regularity

Abstract

We study the stochastic Burgers equation driven by a multiplicative Rosenblatt noise with Hurst parameter . Using a fixed-point argument in a Malliavin--Sobolev space that controls the solution and its first two Malliavin derivatives, we prove local existence and uniqueness of a mild solution. We establish uniform moment bounds of all orders and prove Hölder regularity: spatial Hölder exponent and temporal Hölder exponent , which are shown to be sharp by a lower bound for the linearized equation. The proof relies on sharp estimates of the heat kernel in the reproducing kernel Hilbert space of the Rosenblatt process, on Meyer's inequalities for moment bounds, and on a careful analysis of the Skorohod integral with respect to the Rosenblatt process. These results provide a rigorous foundation for the study of nonlinear SPDEs driven by non-Gaussian long-memory noise.
Paper Structure (47 sections, 30 theorems, 204 equations, 1 table)

This paper contains 47 sections, 30 theorems, 204 equations, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The Rosenblatt process
  4. Malliavin calculus for the Rosenblatt process
  5. Heat kernel estimates
  6. Function spaces and decomposition of the mild solution
  7. Main assumptions
  8. Heat kernel estimates in $\mathcal{H}$
  9. Explicit computation of the $\mathcal{H}$-norm of the heat kernel
  10. Time and space differences
  11. Algebra property of $\mathcal{H}$ under convolution
  12. A priori estimates
  13. Estimates for $\Phi^u$
  14. Estimates for the stochastic convolution
  15. Step 1: Decomposition of $\mathrm{D}_z F$.
  16. ...and 32 more sections

Key Result

Proposition 2.4

Let $\Phi \in \mathbb{D}^{2,2}(L^2([0,T]))$. Then where $\nabla_H$ and $\nabla_H^2$ are fractional derivative operators of order $H$ and $2H$ respectively, defined by with $c_H^R$ a normalization constant (see Table tab:constants).

Theorems & Definitions (69)

  • Remark 2.1
  • Remark 2.2
  • Definition 2.3: Skorohod integral
  • Proposition 2.4: Skorohod isometry, Coupek2022
  • Remark 2.5
  • Lemma 2.6: Basic heat kernel estimates
  • Remark 2.7
  • Remark 2.9
  • Lemma 3.1
  • proof
  • ...and 59 more