Large deviation principles for the stationary solutions and invariant measures of a class of SPDE with locally monotone coefficients
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866915955107430400 |
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| author | Liu, Yong Tang, Bin Zhang, Rangrang |
| author_facet | Liu, Yong Tang, Bin Zhang, Rangrang |
| contents | We establish the well-posedness of stationary solutions for a class of SPDEs with locally monotone coefficients, and prove the Freidlin--Wentzell large deviation principle (LDP) for these stationary solutions. The LDP for the associated invariant measures then follows via the contraction principle, avoiding the need to construct the quasi-potential and verify the Dembo--Zeitouni uniform LDP over bounded sets. By working directly with stationary solutions, we bypass these technical difficulties, thereby providing a more general and flexible framework that is adapted to additive noise, multiplicative noise, and transport-type noise. As applications, our results cover a range of SPDEs, including the stochastic reaction-diffusion equations, stochastic 1D viscous Burgers equation, stochastic 2D Navier--Stokes equations, stochastic 2D magneto-hydrodynamic equations and stochastic 3D hyper-dissipative Navier--Stokes equations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_22461 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Large deviation principles for the stationary solutions and invariant measures of a class of SPDE with locally monotone coefficients Liu, Yong Tang, Bin Zhang, Rangrang Probability 60F10, 60H15, 60J25, 35B40 We establish the well-posedness of stationary solutions for a class of SPDEs with locally monotone coefficients, and prove the Freidlin--Wentzell large deviation principle (LDP) for these stationary solutions. The LDP for the associated invariant measures then follows via the contraction principle, avoiding the need to construct the quasi-potential and verify the Dembo--Zeitouni uniform LDP over bounded sets. By working directly with stationary solutions, we bypass these technical difficulties, thereby providing a more general and flexible framework that is adapted to additive noise, multiplicative noise, and transport-type noise. As applications, our results cover a range of SPDEs, including the stochastic reaction-diffusion equations, stochastic 1D viscous Burgers equation, stochastic 2D Navier--Stokes equations, stochastic 2D magneto-hydrodynamic equations and stochastic 3D hyper-dissipative Navier--Stokes equations. |
| title | Large deviation principles for the stationary solutions and invariant measures of a class of SPDE with locally monotone coefficients |
| topic | Probability 60F10, 60H15, 60J25, 35B40 |
| url | https://arxiv.org/abs/2604.22461 |