The Gaussian structure of a perturbed KPZ

We study the KPZ equation on a circle with an additive spatial perturbation ${\mathrm{\partial }}_{t}h=\frac{1}{2}\mathrm{\Delta }h+\frac{1}{2}\mathrm{\mid }\mathrm{\nabla }h{\mathrm{\mid }}^2+\xi +V$, where $\xi$ is a spacetime white noise and $V$ is a smooth spatial function. When $V=0$, it is well-known that the unique invariant measure is the Brownian bridge. In the presence of the perturbation, we show that the equation admits a unique invariant measure that is absolutely continuous with respect to the Brownian bridge. We further prove the measure has a finite relative entropy with respect to the law of the bridge and that, for any $p\in (1,\mathrm{\infty })$, the corresponding Radon-Nikodym derivative belongs to $L^p$, provided that $\int V^2$ is sufficiently small. The proof uses the discretization and mollification scheme of \cite{FQ}, together with an application of the log-Sobolev and spectral gap inequalities for the underlying Gaussian measure.