Geodesic
On large deviations for ensembles of distributions
Sbornik. Mathematics, Tome 204 (2013) no. 11, pp. 1671-1690 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

The paper is concerned with the large deviations problem in the Freidlin-Wentzell formulation without the assumption of the uniqueness of the solution to the equation involving white noise. In other words, it is assumed that for each $\varepsilon>0$ the nonempty set $\mathscr P_\varepsilon$ of weak solutions is not necessarily a singleton. Analogues of a number of concepts in the theory of large deviations are introduced for the set $\{\mathscr P_\varepsilon,\,\varepsilon>0\}$, hereafter referred to as an ensemble of distributions. The ensembles of weak solutions of an $n$-dimensional stochastic Navier-Stokes system and stochastic wave equation with power-law nonlinearity are shown to be uniformly exponentially tight. An idempotent Wiener process in a Hilbert space and idempotent partial differential equations are defined. The accumulation points in the sense of large deviations of the ensembles in question are shown to be weak solutions of the corresponding idempotent equations. Bibliography: 14 titles.
Keywords: large deviations, $n$-dimensional Navier-Stokes system, nonlinear wave equation. 
@article{SM_2013_204_11_a6,
     author = {D. A. Khrychev},
     title = {On large deviations for ensembles of distributions},
     journal = {Sbornik. Mathematics},
     pages = {1671--1690},
     year = {2013},
     volume = {204},
     number = {11},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_2013_204_11_a6/}
}
TY  - JOUR
AU  - D. A. Khrychev
TI  - On large deviations for ensembles of distributions
JO  - Sbornik. Mathematics
PY  - 2013
SP  - 1671
EP  - 1690
VL  - 204
IS  - 11
UR  - http://geodesic.mathdoc.fr/item/SM_2013_204_11_a6/
LA  - en
ID  - SM_2013_204_11_a6
ER  - 
%0 Journal Article
%A D. A. Khrychev
%T On large deviations for ensembles of distributions
%J Sbornik. Mathematics
%D 2013
%P 1671-1690
%V 204
%N 11
%U http://geodesic.mathdoc.fr/item/SM_2013_204_11_a6/
%G en
%F SM_2013_204_11_a6
D. A. Khrychev. On large deviations for ensembles of distributions. Sbornik. Mathematics, Tome 204 (2013) no. 11, pp. 1671-1690. http://geodesic.mathdoc.fr/item/SM_2013_204_11_a6/

[1] S. S. Sritharan, P. Sundar, “Large deviations for the two-dimensional Navier–Stokes equations with multiplicative noise”, Stochastic Process. Appl., 116:11 (2006), 1636–1659 | DOI | MR | Zbl

[2] A. Budhiraja, P. Dupuis, “A variational representation for positive functionals of infinite dimensional Brownian motion”, Probab. Math. Statist., 20:1 (2000), 39–61 | MR | Zbl

[3] U. Manna, S. S. Sritharan, P. Sundar, “Large deviations for the stochastic shell model of turbulence”, NoDEA Nonlinear Differential Equations Appl., 16:4 (2009), 493–521 | DOI | MR | Zbl

[4] J. Duan, A. Millet, “Large deviations for the Boussinesq equations under random influences”, Stochastic Process. Appl., 119:6 (2009), 2052–2081 | DOI | MR | Zbl

[5] V. Ortiz-López, M. Sanz-Solé, “A Laplace principle for a stochastic wave equation in spatial dimension three”, Stochastic analysis 2010, Springer-Verlag, Heidelberg, 2011, 31–49 | MR | Zbl

[6] M.-H. Chang, “Large deviation for Navier–Stokes equations with small stochastic perturbation”, Appl. Math. Comput., 76:1 (1996), 65–93 | DOI | MR | Zbl

[7] A. A. Pukhalskii, Bolshie ukloneniya stokhasticheskikh dinamicheskikh sistem. Teoriya i prilozheniya, Fizmatlit, M., 2005

[8] J. Feng, T. G. Kurtz, Large deviations for stochastic processes, Math. Surveys Monogr., 131, Amer. Math. Soc., Providence, RI, 2006 | MR | Zbl

[9] P. Marin-Rubio, J. C. Robinson, “Attractors for the stochastic 3D Navier–Stokes equations”, Stoch. Dyn., 3:3 (2003), 279–297 | DOI | MR | Zbl

[10] C. Odasso, “Exponential mixing for stochastic PDEs: the non-additive case”, Probab. Theory Related Fields, 140:1–2 (2008), 41–82 | DOI | MR | Zbl

[11] M. I. Vishik, A. I. Komech, A. V. Fursikov, “Some mathematical problems of statistical hyromechanics”, Russian Math. Surveys, 34:5 (1979), 149–234 | DOI | MR | Zbl

[12] D. A. Hryčev, “On a certain stochastic quasilinear hyperbolic equation”, Math. USSR-Sb., 44:3 (1983), 363–388 | DOI | MR | Zbl | Zbl

[13] D. A. Khrychev, “Optimal programmed controls: existence and approximation”, Sb. Math., 192:5 (2001), 763–783 | DOI | DOI | MR | Zbl

[14] J.-L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires, Dunod, de Gruyter, Paris, 1969 | MR | MR | Zbl | Zbl