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Ergodic and mixing properties of the 2D Navier–Stokes equations with a degenerate multiplicative Gaussian noise

Zhe Dong & Xuhui Peng

Annals of Applied Probability2026https://doi.org/10.1214/25-aap2264article
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Abstract

In this paper, we establish the ergodic and mixing properties of stochastic 2D Navier–Stokes equations driven by a highly degenerate multiplicative Gaussian noise. The noise can appear in as few as four directions, and its intensity depends on the solution. The case of additive Gaussian noise was previously treated by Hairer and Mattingly (Ann. of Math. (2) 164 (2006) 993–1032). To derive the ergodic and mixing properties in the present setting, we employ Malliavin calculus to establish the asymptotically strong Feller property. The primary challenge lies in proving the “invertibility” of the Malliavin matrix, which differs fundamentally from the additive case.

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https://doi.org/https://doi.org/10.1214/25-aap2264

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@article{zhe2026,
  title        = {{Ergodic and mixing properties of the 2D Navier–Stokes equations with a degenerate multiplicative Gaussian noise}},
  author       = {Zhe Dong & Xuhui Peng},
  journal      = {Annals of Applied Probability},
  year         = {2026},
  doi          = {https://doi.org/https://doi.org/10.1214/25-aap2264},
}

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Ergodic and mixing properties of the 2D Navier–Stokes equations with a degenerate multiplicative Gaussian noise

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