Metadata
Abstract
In this paper, we consider the 2D Navier–Stokes system driven by a white-in-time noise. We show that the occupation measures of the trajectories satisfy a large deviations principle, provided that the noise acts directly on all Fourier modes. The proofs are obtained by developing an approach introduced previously for discrete-time random dynamical systems, based on a Kifer-type criterion and a multiplicative ergodic theorem.
References
[BKL02] Exponential mixing of the 2D stochastic Navier–Stokes dynamics, Commun. Math. Phys., Volume 230 (2002) no. 1, pp. 87-132 | Zbl 1033.76011
[DS89] Large Deviations, Academic Press Inc., Pure and Applied Mathematics, Volume 137 (1989) | Zbl 0705.60029
[Dud02] Real analysis and probability, Cambridge University Press, Cambridge Studies in Advanced Mathematics, Volume 74 (2002) | Zbl 1023.60001
[DV75] Asymptotic evaluation of certain Markov process expectations for large time. I–II, Commun. Pure Appl. Math., Volume 28 (1975) no. 1, p. 1-47 & 279–301 | Zbl 0323.60069
[DZ00] Large deviations Techniques and applications, Springer (2000)
[EMS01] Gibbsian dynamics and ergodicity for the stochastically forced Navier–Stokes equation, Commun. Math. Phys., Volume 224 (2001) no. 1, pp. 83-106 | Zbl 0994.60065
[FM95] Ergodicity of the 2D Navier–Stokes equation under random perturbations, Commun. Math. Phys., Volume 172 (1995) no. 1, pp. 119-141 | Zbl 0845.35080
[FP67] Sur le comportement global des solutions non-stationnaires des équations de Navier–Stokes en dimension , Rend. Semin. Mat. Univ. Padova, Volume 39 (1967), pp. 1-34 | Zbl 0176.54103
[FW84] Random perturbations of dynamical systems, Springer, Grundlehren der Mathematischen Wissenschaften, Volume 260 (1984) | Zbl 0522.60055
[Gou07a] A large deviation principle for 2D stochastic Navier–Stokes equation, Stochastic Processes Appl., Volume 117 (2007) no. 7, pp. 904-927 | Zbl 1117.60027
[Gou07b] Large deviation principle of occupation measure for a stochastic Burgers equation, Ann. Inst. Henri Poincaré, Probab. Stat., Volume 43 (2007) no. 4, pp. 375-408
[HM06] Ergodicity of the 2D Navier–Stokes equations with degenerate stochastic forcing, Ann. Math., Volume 164 (2006) no. 3, pp. 993-1032 | Zbl 1130.37038
[JNPS15] Large deviations from a stationary measure for a class of dissipative PDEs with random kicks, Commun. Pure Appl. Math., Volume 68 (2015) no. 12, pp. 2108-2143 | Zbl 1328.60076
[JNPS18] Large deviations and mixing for dissipative PDE’s with unbounded random kicks, Nonlinearity, Volume 31 (2018) no. 2, pp. 540-596 | Zbl 1382.35191
[KNS18] Exponential mixing for a class of dissipative PDEs with bounded degenerate noise (2018) https://arxiv.org/abs/1802.03250
[KS91] Brownian motion and stochastic calculus, Springer, Graduate Texts in Mathematics, Volume 113 (1991) | Zbl 0734.60060
[KS00] Stochastic dissipative PDEs and Gibbs measures, Commun. Math. Phys., Volume 213 (2000) no. 2, pp. 291-330 | Zbl 0974.60046
[KS02] Coupling approach to white-forced nonlinear PDEs, J. Math. Pures Appl., Volume 81 (2002) no. 6, pp. 567-602 | Zbl 1021.37044
[KS12] Mathematics of two-dimensional turbulence, Cambridge University Press, Cambridge Tracts in Mathematics, Volume 194 (2012) | Zbl 1333.76003
[Kuk02] Ergodic theorems for 2D statistical hydrodynamics, Rev. Math. Phys., Volume 14 (2002) no. 6, pp. 585-600 | Zbl 1030.37054
[Lio69] Quelques méthodes de résolution des problèmes aux limites non linéaires, Dunod, Études mathématiques (1969) | Zbl 0189.40603
[MN18a] Local large deviations principle for occupation measures of the stochastic damped nonlinear wave equation, Ann. Inst. Henri Poincaré, Probab. Stat., Volume 54 (2018) no. 4, pp. 2002-2041 | Zbl 06996557
[MN18b] Multiplicative ergodic theorem for a non-irreducible random dynamical system (2018) https://www.archives-ouvertes.fr/hal-01695046v1
[Oda08] Exponential mixing for stochastic PDEs: the non-additive case, Probab. Theory Relat. Fields, Volume 140 (2008) no. 1-2, pp. 41-82 | Zbl 1137.60030
[Shi06] Law of large numbers and central limit theorem for randomly forced PDE’s, Probab. Theory Relat. Fields, Volume 134 (2006) no. 2, pp. 215-247 | Zbl 1099.35188
[Wu01] Large and moderate deviations and exponential convergence for stochastic damping Hamiltonian systems, Stochastic Processes Appl., Volume 91 (2001) no. 2, pp. 205-238 | Zbl 1047.60059