Abstract
Exponential integrability of the solution to the stochastic Burgers equation driven by white noise
We study the 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
\[
\mathbb{E}
\left[
\exp\left(
\lambda \sup_{t \in [0,T]}
\|X_t^x\|_{L^2(0,1)}^2
\right)
\right]
< \infty.
\]
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 result, we combine the Boué–Dupuis method with an argument used in Da Prato–Debussche, Potential Analysis, 2007.
These estimates have interesting applications. We also deduce a new Lipschitz regularizing effect for the corresponding Markov semigroup.
This is joint work with Francesco C. De Vecchi and Josef Janák.