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The ergodic theory of SPDEs in a weak-noise regime

Probability 2026-03-20 v1

Abstract

Consider a parabolic SPDE tu=Δu+σ(u)η, \partial_t u = \Delta u + \sigma(u)\eta, on (0,)×Rd(0\,,\infty)\times\mathbb{R}^d, where η\eta is a centered, generalized Gaussian noise with Cov[η(t,x),η(s,y)]=δ0(ts)Λ(xy)\text{Cov}[\eta(t\,,x)\,,\eta(s\,,y)]=\delta_0(t-s)\Lambda(x-y) for a tempered Borel measure Λ\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.

Keywords

Cite

@article{arxiv.2603.18384,
  title  = {The ergodic theory of SPDEs in a weak-noise regime},
  author = {Mathew Joseph and Davar Khoshnevisan and Kunwoo Kim and Carl Mueller},
  journal= {arXiv preprint arXiv:2603.18384},
  year   = {2026}
}

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49 pages