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Quasi-Gaussianity of the 2D stochastic Navier-Stokes equations

James Coe, Martin Hairer, Leonardo Tolomeo

TL;DR

This work analyzes the long-time behavior of the 2D stochastic Navier–Stokes equations on $oldsymbol{T}^2$ with additive, spatially colored forcing and proves that the unique invariant measure $ ho$ is equivalent to the Gaussian invariant $oldsymbol{ ho}_ ext{OU}$ of the linear equation. The authors develop a generalized time-shifted Girsanov framework and an abstract equivalence criterion to compare nonlinear SPDEs via Cameron–Martin shifts, relying on a linearised propagator $J$ and a careful handling of regularity. Under the regime $oldsymbol{ heta}>2/3$ and $oldsymbol{eta}>2-oldsymbol{ heta}$ (with a corollary extending to the $oldsymbol{ heta}=1$ case for $oldsymbol{eta}>0$), the nonlinear NS system and its twisted Gaussian-invariant counterpart share mutually absolutely continuous time marginals, and their invariant measures are equivalent. The approach also extends to hypoviscous variants, highlighting a robust quasi-Gaussian structure in 2D stochastic NSE and enabling Gaussian-style moment and ergodic analysis. Overall, the paper provides a rigorous bridge between nonlinear SPDE dynamics and Gaussian baselines through a refined Girsanov-based coupling and harmonic-analysis tools.

Abstract

We study the qualitative properties of solutions to the 2D stochastic Navier-Stokes equations with forcing that is white in time and coloured in space. Our main result shows that the unique invariant measure of this system is equivalent to that of the corresponding Ornstein-Uhlenbeck process. Our method relies on a generalization of the "time-shifted Girsanov method" of [MS05, MRS22] to compare the laws of time marginals for dissipative SPDEs. This generalisation allows to not only compare solutions to a nonlinear equation to those of the corresponding linear equation, but also to directly compare two nonlinear equations. We use this to establish equivalence of the Navier-Stokes system to a "twisted" nonlinear system that leaves the Gaussian measure invariant. We further apply this method to establish similar equivalence statements for a family of hypoviscous Navier-Stokes equations.

Quasi-Gaussianity of the 2D stochastic Navier-Stokes equations

TL;DR

This work analyzes the long-time behavior of the 2D stochastic Navier–Stokes equations on with additive, spatially colored forcing and proves that the unique invariant measure is equivalent to the Gaussian invariant of the linear equation. The authors develop a generalized time-shifted Girsanov framework and an abstract equivalence criterion to compare nonlinear SPDEs via Cameron–Martin shifts, relying on a linearised propagator and a careful handling of regularity. Under the regime and (with a corollary extending to the case for ), the nonlinear NS system and its twisted Gaussian-invariant counterpart share mutually absolutely continuous time marginals, and their invariant measures are equivalent. The approach also extends to hypoviscous variants, highlighting a robust quasi-Gaussian structure in 2D stochastic NSE and enabling Gaussian-style moment and ergodic analysis. Overall, the paper provides a rigorous bridge between nonlinear SPDE dynamics and Gaussian baselines through a refined Girsanov-based coupling and harmonic-analysis tools.

Abstract

We study the qualitative properties of solutions to the 2D stochastic Navier-Stokes equations with forcing that is white in time and coloured in space. Our main result shows that the unique invariant measure of this system is equivalent to that of the corresponding Ornstein-Uhlenbeck process. Our method relies on a generalization of the "time-shifted Girsanov method" of [MS05, MRS22] to compare the laws of time marginals for dissipative SPDEs. This generalisation allows to not only compare solutions to a nonlinear equation to those of the corresponding linear equation, but also to directly compare two nonlinear equations. We use this to establish equivalence of the Navier-Stokes system to a "twisted" nonlinear system that leaves the Gaussian measure invariant. We further apply this method to establish similar equivalence statements for a family of hypoviscous Navier-Stokes equations.
Paper Structure (10 sections, 16 theorems, 49 equations)

This paper contains 10 sections, 16 theorems, 49 equations.

Table of Contents

  1. Introduction
  2. Notations
  3. Idea of proof
  4. Properties of the solution to stochastic Navier--Stokes equations
  5. Bounds on stochastic convolution
  6. Linearised equation
  7. Strong Feller property
  8. Proof of the main theorems
  9. An abstract equivalence criterion
  10. Application to stochastic Navier--Stokes equations

Key Result

Theorem 1.1

For every $\alpha > 0$, e:SNS admits a unique invariant measure $\rho$, which is equivalent to $\mu_\alpha$.

Theorems & Definitions (38)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.3
  • Remark 1.4
  • Remark 1.5
  • Proposition 2.1
  • proof
  • Remark 2.2
  • Remark 2.3
  • Lemma 3.1
  • ...and 28 more