Exponential integrability of the solution to the stochastic Burgers equation driven by white noise

Abstract

We study stochastic Burgers equation driven by a rough noise $(-\Delta {)}^{\gamma }d{W}_t$, where $\Delta$ is the Laplacian in one dimension with Dirichlet boundary conditions, and $\gamma \in [0,1/4)$. 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 $\lambda >0$ 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 $-1/2<\gamma <-1/4$ 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.