The ergodic theory of SPDEs in a weak-noise regime

TL;DR

This paper characterizes all ergodic invariant measures for a class of parabolic SPDEs driven by Gaussian noise in the weak-noise regime, advancing understanding of their long-term statistical behavior.

Contribution

It provides a complete classification of ergodic invariant measures for these SPDEs under weak-noise conditions, a significant extension of prior existence results.

Findings

All ergodic invariant measures are characterized.

The results apply to a broad class of Gaussian noise measures.

The work advances understanding of long-term behavior of SPDEs.

Abstract

Consider a parabolic SPDE \[ \partial_t u = \Delta u + \sigma(u)\eta, \] on $(0\,,\infty)\times\mathbb{R}^d$, where $\eta$ is a centered, generalized Gaussian noise with $\text{Cov}[\eta(t\,,x)\,,\eta(s\,,y)]=\delta_0(t-s)\Lambda(x-y)$ for a tempered Borel measure $\Lambda$ 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.