The ergodic theory of SPDEs in a weak-noise regime

Consider a parabolic SPDE $${\mathrm{\partial }}_{t}u=\mathrm{\Delta }u+\sigma (u)\eta ,$$ on $(0,\mathrm{\infty })\times {\mathbb{R}}^d$, where $\eta$ is a centered, generalized Gaussian noise with $\text{Cov}[\eta (t,x),\eta (s,y)]={\delta }_0(t-s)\mathrm{\Lambda }(x-y)$ for a tempered Borel measure $\mathrm{\Lambda }$ that is positive definite and satisfies a mild weak-noise. The existence of invariant measures of versions of these types of SPDEs has been studied at great length, particularly in the ``weak-noise regime’'; see for example Assing and Manthey \cite{AssingManthey2003}, Chen and Eisenberg \cite{ChenEisenberg2024}, Chen, Ouyang, Tindel, and Xia \cite{ChenOuyangTindelXia2024}, Eckmann and Hairer \cite{EckmannHairer2001}, Misiats and Stanzhytskyi \cite{MSY2020}, Yu Gu and Jiawei Li \cite{GuLi2020}, and Tessitore and Zabczyk \cite{TessitoreZabczyk1998}. Here, we characterize all annealed, ergodic, invariant measures for the above SPDE in the weak-noise regime.