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

TL;DR

This paper establishes exponential integrability estimates for solutions to a stochastic Burgers equation driven by rough noise, extending known results to a broader noise regime and employing advanced probabilistic techniques.

Contribution

It proves exponential estimates for solutions driven by rough noise, combining Boué-Dupuis method with Da Prato-Debussche argument, expanding applicability beyond trace class noise.

Findings

Exponential integrability holds for solutions driven by noise with b3 a7 [0,1/4)

The method extends exponential estimates beyond trace class noise cases

Results have applications in large deviation theory and regularizing effects

Abstract

We study stochastic Burgers equation driven by a rough noise $(-\Delta)^{\gamma} dW_t$, 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(\lambda \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 It\^o formula. To prove the exponential estimate we combine the Bou\'e-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.