Speaker
Description
We study stochastic Burgers equation driven by a rough noise $(-\Delta)^{\gamma} \: \mathrm d W_t$, 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}\tag{$\star$}
\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 result we combine the Bou\'e-Dupuis method with an argument used in [Da Prato-Debussche, Potential Anal. 2007]. Estimate \eqref{ds} has interesting applications. We also deduce a new Lipschitz regularizing effect for the corresponding Markov semigroup. This is a joint work with Francesco C. De Vecchi and Josef Janák.