Exponential integrability of the solution to the stochastic Burgers equation driven by white noise
Fuente:
arXiv
Salvato in:
| Autori principali: | , , |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2026
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866916054981148672 |
|---|---|
| author | De Vecchi, Francesco C. Janák, Josef Priola, Enrico |
| author_facet | De Vecchi, Francesco C. Janák, Josef Priola, Enrico |
| contents | We study stochastic Burgers equation driven by a rough noise $(-Δ)^γ dW_t$, where $Δ$ is the Laplacian in one dimension with Dirichlet boundary conditions, and $γ\in [0,1/4)$. We prove exponential estimates for the solution $X_t^x$, starting from $x \in L^2(0,1)$, by showing that there exists some constant $λ>0$ for which \begin{equation} \label{ds} \mathbb{E}
\left[\exp\left(λ\sup_{t\in[0,T]}\|X_t^x\|_{L^2(0,1)}^2 \right) \right]< \infty. \end{equation} This estimate was known only in the case of trace class noise when $-1/2 <γ< -1/4 $ since in that case one can use the Itô formula. To prove the exponential estimate we combine the Boué-Dupuis method with an argument used in [Da Prato-Debussche, Potential Anal. 2007]. The exponential estimate have important applications in large deviation theory, among others. We also deduce a new Lipschitz regularizing effect for the corresponding Markov semigroup. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_03182 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Exponential integrability of the solution to the stochastic Burgers equation driven by white noise De Vecchi, Francesco C. Janák, Josef Priola, Enrico Probability 60H15, 35R60 We study stochastic Burgers equation driven by a rough noise $(-Δ)^γ dW_t$, where $Δ$ is the Laplacian in one dimension with Dirichlet boundary conditions, and $γ\in [0,1/4)$. We prove exponential estimates for the solution $X_t^x$, starting from $x \in L^2(0,1)$, by showing that there exists some constant $λ>0$ for which \begin{equation} \label{ds} \mathbb{E} \left[\exp\left(λ\sup_{t\in[0,T]}\|X_t^x\|_{L^2(0,1)}^2 \right) \right]< \infty. \end{equation} This estimate was known only in the case of trace class noise when $-1/2 <γ< -1/4 $ since in that case one can use the Itô formula. To prove the exponential estimate we combine the Boué-Dupuis method with an argument used in [Da Prato-Debussche, Potential Anal. 2007]. The exponential estimate have important applications in large deviation theory, among others. We also deduce a new Lipschitz regularizing effect for the corresponding Markov semigroup. |
| title | Exponential integrability of the solution to the stochastic Burgers equation driven by white noise |
| topic | Probability 60H15, 35R60 |
| url | https://arxiv.org/abs/2605.03182 |