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The ergodic theory of SPDEs in a weak-noise regime

Mathew Joseph, Davar Khoshnevisan, Kunwoo Kim, Carl Mueller

Abstract

Consider a parabolic SPDE \[ \partial_t u = Δu + σ(u)η, \] on $(0\,,\infty)\times\mathbb{R}^d$, where $η$ is a centered, generalized Gaussian noise with $\text{Cov}[η(t\,,x)\,,η(s\,,y)]=δ_0(t-s)Λ(x-y)$ for a tempered Borel measure $Λ$ 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.

The ergodic theory of SPDEs in a weak-noise regime

Abstract

Consider a parabolic SPDE on , where is a centered, generalized Gaussian noise with for a tempered Borel measure 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.
Paper Structure (15 sections, 25 theorems, 228 equations)

This paper contains 15 sections, 25 theorems, 228 equations.

Table of Contents

  1. Introduction
  2. Background information
  3. Duality and invariance
  4. Annealing, and the main result
  5. On Conjecture \ref{['conj2']}
  6. Proof of Theorem \ref{['th:main']}
  7. Stability
  8. Proof of Theorem \ref{['th:main']}(a)
  9. Proof of Theorem \ref{['th:main']}(b)
  10. Proof of Theorem \ref{['th:main']}(c)
  11. Proof of Theorem \ref{['th:main']}(d)
  12. Hölder continuity
  13. The constant coefficient case
  14. The Parabolic Anderson Model
  15. Spatial ergodicity

Key Result

Corollary 1.2

Choose and fix an arbitrary $\theta\in\mathbb{R}$. Then, $\nu_\theta$ is the only mean-$\theta$ time-ergodic invariant measure among all laws of spatially stationary, spatially ergodic random fields should there be any.

Theorems & Definitions (56)

  • Corollary 1.2
  • Conjecture 1
  • Conjecture 2
  • Definition 2.1
  • Proposition 2.2
  • Lemma 2.3
  • Remark 2.4
  • proof
  • Lemma 2.5
  • proof
  • ...and 46 more