Stochastic Burgers equation driven by multiplicative Rosenblatt noise: local existence, uniqueness and regularity
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866912907181162496 |
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| author | Lechiheb, Atef |
| author_facet | Lechiheb, Atef |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_14617 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Stochastic Burgers equation driven by multiplicative Rosenblatt noise: local existence, uniqueness and regularity Lechiheb, Atef Probability Primary 60H15, Secondary 60G22, 60H07, 35R60, 35Q53 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. |
| title | Stochastic Burgers equation driven by multiplicative Rosenblatt noise: local existence, uniqueness and regularity |
| topic | Probability Primary 60H15, Secondary 60G22, 60H07, 35R60, 35Q53 |
| url | https://arxiv.org/abs/2602.14617 |