Ergodicity for 2D Navier-Stokes equations with a degenerate pure jump noise
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| Main Authors: | , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866916230521159680 |
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| author | Peng, Xuhui Zhai, Jianliang Zhang, Tusheng |
| author_facet | Peng, Xuhui Zhai, Jianliang Zhang, Tusheng |
| contents | In this paper, we establish the ergodicity for stochastic 2D Navier-Stokes equations driven by a highly degenerate pure jump Lévy 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_00414 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Ergodicity for 2D Navier-Stokes equations with a degenerate pure jump noise Peng, Xuhui Zhai, Jianliang Zhang, Tusheng Probability In this paper, we establish the ergodicity for stochastic 2D Navier-Stokes equations driven by a highly degenerate pure jump Lévy 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. |
| title | Ergodicity for 2D Navier-Stokes equations with a degenerate pure jump noise |
| topic | Probability |
| url | https://arxiv.org/abs/2405.00414 |