Large deviations principle for invariant measures of stochastic Burgers equations

TL;DR

This paper establishes a large deviations principle for solutions and invariant measures of stochastic Burgers equations under small noise asymptotics, including cases with highly singular noise and specific covariance operator limits.

Contribution

It extends large deviations analysis to stochastic Burgers equations with singular noise and invariant measures, covering new covariance operator limits and uniform initial value considerations.

Findings

Large deviations principle for solutions with singular noise.

Large deviations principle for invariant measures.

Results hold uniformly over initial conditions.

Abstract

We study the small noise asymptotic for stochastic Burgers equations on $(0,1)$ with Dirichlet boundary condition. We consider the case that the noise is more singular than space-time white noise. We let the noise magnitude $\sqrt{\epsilon} \rightarrow 0$ and the covariance operator $Q_\epsilon$ is convergent to $(-\Delta)^{\frac 1 2}$ and prove a large deviations principle for solutions, uniformly with respect to the initial value of equation. Furthermore, we set $Q_\epsilon$ to be a trace class operator and converge to $(-\Delta)^{\frac{\alpha}{2}}$ with $\alpha<1$ in a suitable way such that the invariant measures exist. Then, we prove the large deviations principle for the invariant measures of stochastic Burgers equations.