On ergodic properties of stochastic PDEs
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866912143914303488 |
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| author | Chen, Le Ouyang, Cheng Tindel, Samy Xia, Panqiu |
| author_facet | Chen, Le Ouyang, Cheng Tindel, Samy Xia, Panqiu |
| contents | In this note we review several situations in which stochastic PDEs exhibit ergodic properties. We begin with the basic dissipative conditions, as stated by Da Prato and Zabczyk in their classical monograph. Then we describe the singular case of SPDEs with reflection. Next we move to some degenerate (and thus more demanding) settings. Namely we recall some results obtained around 2006, concerning stochastic Navier-Stokes equations with a very degenerate noise. We finish the article by handling some cases with degenerate coefficients. This includes a new result about the parabolic Anderson model in dimension $d\ge 3$, driven by a general class of noises and fairly general initial conditions. In this context, a phase transition is observed, expressed in terms of the noise intensity. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_03521 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On ergodic properties of stochastic PDEs Chen, Le Ouyang, Cheng Tindel, Samy Xia, Panqiu Probability 60H15, 35R60, 37L40, 35K05 In this note we review several situations in which stochastic PDEs exhibit ergodic properties. We begin with the basic dissipative conditions, as stated by Da Prato and Zabczyk in their classical monograph. Then we describe the singular case of SPDEs with reflection. Next we move to some degenerate (and thus more demanding) settings. Namely we recall some results obtained around 2006, concerning stochastic Navier-Stokes equations with a very degenerate noise. We finish the article by handling some cases with degenerate coefficients. This includes a new result about the parabolic Anderson model in dimension $d\ge 3$, driven by a general class of noises and fairly general initial conditions. In this context, a phase transition is observed, expressed in terms of the noise intensity. |
| title | On ergodic properties of stochastic PDEs |
| topic | Probability 60H15, 35R60, 37L40, 35K05 |
| url | https://arxiv.org/abs/2412.03521 |