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

2024/12/23 by Zhao Dong, Dong, Zhao, Xuhui Peng +1 · 1 citation
Economics, Econometrics and Finance · Engineering · Mathematics · #FOS: Mathematics #Navier-Stokes equation solutions #Probability (math.PR) #Stability and Controllability of Differential Equations #Stochastic processes and financial applications

paper · pdf · doi:10.48550/arxiv.2412.17380

openalex publication_date 2024/12/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01

Abstract

In this paper, we establish ergodic and mixing properties of stochastic 2D Navier-Stokes equations driven by a highly degenerate multiplicative Gaussian noise. The noise could appear in as few as four directions and the intensity of the noise depends on the solution. The case of additive Gaussian noise was treated in Hairer and Mattingly [Ann. of Math., 164(3):993--1032, 2006]. To obtain ergodic and mixing properties, we use Malliavin calculus to establish the asymptotically strong Feller property. The main difficulty lies in the proof of the "invertibility" of Malliavin matrix which is totally different from the additive case.

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• Asymptotic stability and mean ergodicity of Feller processes on Polish spaces

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