Nersesyan, Vahagn
Large deviations for the Navier–Stokes equations driven by a white-in-time noise
Annales Henri Lebesgue, Volume 2 (2019) , pp. 481-513.


KeywordsStochastic Navier–Stokes system, large deviations principle, occupation measures, multiplicative ergodicity


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.


[BKL02] Bricmont, Jean; Kupiainen, Antti; Lefevere, Raphaël Exponential mixing of the 2D stochastic Navier–Stokes dynamics, Commun. Math. Phys., Volume 230 (2002) no. 1, pp. 87-132 | Zbl 1033.76011

[DS89] Deuschel, Jean-Dominique; Stroock, Daniel W. Large Deviations, Academic Press Inc., Pure and Applied Mathematics, Volume 137 (1989) | Zbl 0705.60029

[Dud02] Dudley, Richard M. Real analysis and probability, Cambridge University Press, Cambridge Studies in Advanced Mathematics, Volume 74 (2002) | Zbl 1023.60001

[DV75] Donsker, Monroe D.; Varadhan, S. R. Srinivasa 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] Dembo, Amir; Zeitouni, Ofer Large deviations Techniques and applications, Springer (2000)

[EMS01] E, Weinan; Mattingly, Jonathan C.; Sinaĭ, Yakov G. 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] Flandoli, Franco; Maslowski, Bohdan 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] Foiaš, Ciprian; Prodi, Giovanni Sur le comportement global des solutions non-stationnaires des équations de Navier–Stokes en dimension 2, Rend. Semin. Mat. Univ. Padova, Volume 39 (1967), pp. 1-34 | Zbl 0176.54103

[FW84] Freidlin, Mark I.; Wentzell, Alexander D. Random perturbations of dynamical systems, Springer, Grundlehren der Mathematischen Wissenschaften, Volume 260 (1984) | Zbl 0522.60055

[Gou07a] Gourcy, Mathieu 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] Gourcy, Mathieu 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] Hairer, Martin; Mattingly, Jonathan C. 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] Jakšić, Vojkan; Nersesyan, Vahagn; Pillet, Claude-Alain; Shirikyan, Armen 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] Jakšić, Vojkan; Nersesyan, Vahagn; Pillet, Claude-Alain; Shirikyan, Armen 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] Kuksin, Sergei; Nersesyan, Vahagn; Shirikyan, Armen Exponential mixing for a class of dissipative PDEs with bounded degenerate noise (2018)

[KS91] Karatzas, Ioannis; Shreve, Steven E. Brownian motion and stochastic calculus, Springer, Graduate Texts in Mathematics, Volume 113 (1991) | Zbl 0734.60060

[KS00] Kuksin, Sergei; Shirikyan, Armen Stochastic dissipative PDEs and Gibbs measures, Commun. Math. Phys., Volume 213 (2000) no. 2, pp. 291-330 | Zbl 0974.60046

[KS02] Kuksin, Sergei; Shirikyan, Armen Coupling approach to white-forced nonlinear PDEs, J. Math. Pures Appl., Volume 81 (2002) no. 6, pp. 567-602 | Zbl 1021.37044

[KS12] Kuksin, Sergei; Shirikyan, Armen Mathematics of two-dimensional turbulence, Cambridge University Press, Cambridge Tracts in Mathematics, Volume 194 (2012) | Zbl 1333.76003

[Kuk02] Kuksin, Sergei B. Ergodic theorems for 2D statistical hydrodynamics, Rev. Math. Phys., Volume 14 (2002) no. 6, pp. 585-600 | Zbl 1030.37054

[Lio69] Lions, Jacques-Louis Quelques méthodes de résolution des problèmes aux limites non linéaires, Dunod, Études mathématiques (1969) | Zbl 0189.40603

[MN18a] Martirosyan, Davit; Nersesyan, Vahagn 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] Martirosyan, Davit; Nersesyan, Vahagn Multiplicative ergodic theorem for a non-irreducible random dynamical system (2018)

[Oda08] Odasso, Cyril 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] Shirikyan, Armen 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] Wu, Liming 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