Weakly nonlinear stochastic CGL equations
Annales de l'I.H.P. Probabilités et statistiques, Tome 49 (2013) no. 4, pp. 1033-1056.

Nous considérons l’équation de Schrödinger linéaire avec les conditions aux limites périodiques, perturbée par une force aléatoire et amortie par un terme quasi linéaire:

d dtu+i- Δ + V ( x )u=νΔ u - γ R |u| 2p u - i γ I |u| 2q u+νη(t,x).(*)
La force η est un processus aléatoire blanc en temps t et lisse en x; le potentiel V(x) est typique. Nous étudions le comportement asymptotique des solutions sur de longs intervalles de temps 0tν -1 T, quand ν0, et le comportement des solutions quand t et ν0. Nous démontrons qu’on peut décrire ces deux comportements asymptotiques en termes des solutions du système d'équations effectives pour (*). Ce dernier est une équation de la chaleur avec un terme quasi linéaire non local et une force aléatoire lisse additive, qui est écrite dans l’espace de Fourier. Les équations ne dépendent pas de la partie hamiltonienne de la perturbation -iγ I |u| 2q u (mais elles dépendent de la partie dissipative -γ R |u| 2p u). Si p est un entier, on peut écrire ces équations explicitement.

We consider the linear Schrödinger equation under periodic boundary conditions, driven by a random force and damped by a quasilinear damping:

d dtu+i- Δ + V ( x )u=νΔ u - γ R |u| 2p u - i γ I |u| 2q u+νη(t,x).(*)
The force η is white in time and smooth in x; the potential V(x) is typical. We are concerned with the limiting, as ν0, behaviour of solutions on long time-intervals 0tν -1 T, and with behaviour of these solutions under the double limit t and ν0. We show that these two limiting behaviours may be described in terms of solutions for the system of effective equations for (*) which is a well posed semilinear stochastic heat equation with a non-local nonlinearity and a smooth additive noise, written in Fourier coefficients. The effective equations do not depend on the Hamiltonian part of the perturbation -iγ I |u| 2q u (but depend on the dissipative part -γ R |u| 2p u). If p is an integer, they may be written explicitly.

DOI : 10.1214/11-AIHP482
Classification : 35Q56, 60H15
Mots-clés : complex Ginzburg-Landau equation, small nonlinearity, stationary measures, averaging, effective equations
@article{AIHPB_2013__49_4_1033_0,
     author = {Kuksin, Sergei B.},
     title = {Weakly nonlinear stochastic {CGL} equations},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     pages = {1033--1056},
     publisher = {Gauthier-Villars},
     volume = {49},
     number = {4},
     year = {2013},
     doi = {10.1214/11-AIHP482},
     mrnumber = {3127912},
     zbl = {1280.35144},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1214/11-AIHP482/}
}
TY  - JOUR
AU  - Kuksin, Sergei B.
TI  - Weakly nonlinear stochastic CGL equations
JO  - Annales de l'I.H.P. Probabilités et statistiques
PY  - 2013
SP  - 1033
EP  - 1056
VL  - 49
IS  - 4
PB  - Gauthier-Villars
UR  - http://archive.numdam.org/articles/10.1214/11-AIHP482/
DO  - 10.1214/11-AIHP482
LA  - en
ID  - AIHPB_2013__49_4_1033_0
ER  - 
%0 Journal Article
%A Kuksin, Sergei B.
%T Weakly nonlinear stochastic CGL equations
%J Annales de l'I.H.P. Probabilités et statistiques
%D 2013
%P 1033-1056
%V 49
%N 4
%I Gauthier-Villars
%U http://archive.numdam.org/articles/10.1214/11-AIHP482/
%R 10.1214/11-AIHP482
%G en
%F AIHPB_2013__49_4_1033_0
Kuksin, Sergei B. Weakly nonlinear stochastic CGL equations. Annales de l'I.H.P. Probabilités et statistiques, Tome 49 (2013) no. 4, pp. 1033-1056. doi : 10.1214/11-AIHP482. http://archive.numdam.org/articles/10.1214/11-AIHP482/

[1] A. Agrachev, S. Kuksin, A. Sarychev and A. Shirikyan. On finite-dimensional projections of distributions for solutions of randomly forced PDEs. Ann. Inst. Henri Poincarè Probab. Stat. 43 (2007) 399-415. | MR

[2] V. Arnold, V. V. Kozlov and A. I. Neistadt. Mathematical Aspects of Classical and Celestial Mechanics, 3rd edition. Springer, Berlin, 2006. | MR

[3] A. Debussche and C. Odasso. Ergodicity for the weakly damped stochastic non-linear Shrödinger equations. J. Evol. Equ. 5 (2005) 317-356. | MR

[4] M. Freidlin and A. Wentzell. Random Perturbations of Dynamical Systems, 2nd edition. Springer, New York, 1998. | MR

[5] M. Hairer. Exponential mixing properties of stochastic PDE's through asymptotic coupling. Probab. Theory Related Fields 124 (2002) 345-380. | MR

[6] T. Kappeler and S. Kuksin. Strong nonresonance of Schrödinger operators and an averaging theorem. Phys. D 86 (1995) 349-362. | MR

[7] I. Karatzas and S. Shreve. Brownian Motion and Stochastic Calculus, 2nd edition. Springer, Berlin, 1991. | MR

[8] R. Khasminski. On the avaraging principle for Ito stochastic differential equations. Kybernetika 4 (1968) 260-279 (in Russian).

[9] S. B. Kuksin. Damped-driven KdV and effective equations for long-time behaviour of its solutions. Geom. Funct. Anal. 20 (2010) 1431-1463. | MR

[10] S. B. Kuksin and A. L. Piatnitski. Khasminskii-Whitham averaging for randomly perturbed KdV equation. J. Math. Pures Appl. 89 (2008) 400-428. | MR

[11] S. B. Kuksin and A. Shirikyan. Stochastic dissipative PDEs and Gibbs measures. Comm. Math. Phys. 213 (2000) 291-330. | MR

[12] S. B. Kuksin and A. Shirikyan. Randomly forced CGL equation: Stationary measures and the inviscid limit. J. Phys. A 37 (2004) 1-18. | MR

[13] S. B. Kuksin and A. Shirikyan. Mathematics of two-dimensional turbulence. Preprint, 2012. Available at www.math.polytechnique.fr/~kuksin/books.html.

[14] P. Lochak and C. Meunier. Multiphase Averaging for Classical Systems. Springer, New York-Berlin-Heidelberg, 1988. | MR

[15] S. Nazarenko. Wave Turbulence. Springer, Berlin, 2011. | MR

[16] C. Odasso. Ergodicity for the stochastic complex Ginzburg-Landau equations. Ann. Inst. Henri Poincaré Probab. Stat. 42 (2006) 417-454. | MR

[17] J. Poschel and E. Trubowitz. Inverse Spectral Theory. Academic Press, Boston, 1987. | MR

[18] A. Shirikyan. Ergodicity for a class of Markov processes and applications to randomly forced PDE's. II. Discrete Contin. Dyn. Syst. 6 (2006) 911-926. | MR

[19] M. Yor. Existence et unicité de diffusion à valeurs dans un espace de Hilbert. Ann. Inst. Henri Poincaré Probab. Stat. 10 (1974) 55-88. | Numdam | MR

Cité par Sources :