Ergodicity for 2D Navier-Stokes equations with a degenerate pure jump noise

Xuhui Peng; Jianliang Zhai; Tusheng Zhang

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arXiv · 2405.00414

Ergodicity for 2D Navier-Stokes equations with a degenerate pure jump noise

Xuhui Peng·Jianliang Zhai·Tusheng Zhang

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Abstract

In this paper, we establish the ergodicity for stochastic 2D Navier-Stokes equations driven by a highly degenerate pure jump L\'evy noise. The noise could appear in as few as four directions. This gives an affirmative anwser to a longstanding problem. The case of Gaussian noise was treated in Hairer and Mattingly [\emph{Ann. of Math.}, 164(3):993--1032, 2006]. To obtain the uniqueness of invariant measure, we use Malliavin calculus and anticipating stochastic calculus to establish the equi-continuity of the semigroup, the so-called {\em e-property}, and prove some weak irreducibility of the solution process.

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Xuhui Peng, Jianliang Zhai, Tusheng Zhang. 2024-05-01. Ergodicity for 2D Navier-Stokes equations with a degenerate pure jump noise. https://arxiv.org/abs/2405.00414

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