. Dans, sous certaines conditions sur ? et f très similaires aux notres, ils ont pu fabriquer un semi-groupe de transition de Markov associéassociéà

. Notre-résultat-signifie-que-le-semi-groupe and . Qu, ils ont construit est exponentiellement mélangeant. Dans [24], un argument de sélection Markovienne a permit d'´ etablir l'existence d'uné evolution Markovienne associéè a (5.0.1) Notre résultat ne s'applique pas directement car nous ne travaillons qu'avec des solutions limites d'approximations de Galerkin. Néanmoins, nous pensons que notre preuve peutêtrepeutêtre adaptée pour prouver que

. Dans-le, exciter 4 modes de NS pour avoir unicité de la mesure invariante. Ici, la difficulté ne se situe pas dans la dégénérescence du bruit. Nous travaillons avec uné equation dont nous ignorons si elle est bien posée ou non. Ce qui va induire des modifications substantielles de nos preuves. Pour cette raison, nous nous contentons de prouver un résultat plus modeste. En l'occurrence, nous nous restreignonsàrestreignonsà des bruits non dégénérés. L'idée principale est de coupler les solutions lorsque les conditions initiales sont petites au sens d'une norme suffisammentrégulì ere. Pour fabriquer ledit couplage, on utilisera une formule de type Bismuth-Elworthy-Li, autre ingrédient important de la preuve est le fait que le temps d'entrée des solutions faibles dans une petite boule admet un moment exponentiel. On compense le manque d'unicité en travaillant avec des approximations de Galerkin et passantàpassantà la limite

M. Barton-smith, Invariant measure for the stochastic Ginzburg Landau equation, Nonlinear Differential Equations Appl, pp.29-52, 2004.

P. Bebouche and A. , Inviscid Limits??of the Complex Ginzburg???Landau Equation, Communications in Mathematical Physics, vol.214, issue.1, pp.201-226, 2000.
DOI : 10.1007/s002200000263

A. Bensoussan, Stochastic Navier-Stokes Equations, Acta Applicandae Mathematicae, vol.38, issue.3, pp.267-304, 1995.
DOI : 10.1007/BF00996149

A. Bensoussan and R. Temam, Equations stochastiques du type Navier-Stokes, Journal of Functional Analysis, vol.13, issue.2, pp.195-222, 1973.
DOI : 10.1016/0022-1236(73)90045-1

J. Bricmont, A. Kupiainen, and R. Lefevere, Exponential mixing for the 2D stochastic Navier-Stokes dynamics, Communications in Mathematical Physics, vol.230, issue.1, pp.87-132, 2002.
DOI : 10.1007/s00220-002-0708-1

M. Capinski and N. Cutland, Statistical solutions of stochastic Navier-Stokes equations, Indiana Univ, Math. J, vol.43, issue.3, pp.927-940, 1994.

M. Capinski and D. Gatarek, Stochastic Equations in Hilbert Space with Application to Navier-Stokes Equations in Any Dimension, Journal of Functional Analysis, vol.126, issue.1, pp.26-35, 1994.
DOI : 10.1006/jfan.1994.1140

G. Da-prato and A. Debussche, Ergodicity for the 3D stochastic Navier???Stokes equations, Journal de Math??matiques Pures et Appliqu??es, vol.82, issue.8, pp.877-947, 2003.
DOI : 10.1016/S0021-7824(03)00025-4

G. Da-prato and J. Zabczyk, Stochastic equations in infinite dimensions , Encyclopedia of Mathematics and its Applications, p.78, 1992.

G. Da-prato and J. Zabczyk, Ergodicity for Infinite Dimensional Systems, 1996.
DOI : 10.1017/CBO9780511662829

A. De-bouard and A. Debussche, A Stochastic Nonlinear Schr??dinger Equation??with Multiplicative Noise, Communications in Mathematical Physics, vol.205, issue.1, pp.161-181, 1999.
DOI : 10.1007/s002200050672

A. De-bouard and A. Debussche, The stochastic non-linear Schrödinger equation in H 1 , Stochastic Analysis and applications 21, pp.197-126, 2003.

A. Debussche and C. Odasso, Ergodicity for the weakly damped stochastic Non-linear Schrödinger equations

W. E. , J. C. Mattingly, and Y. G. Sinai, Gibbsian dynamics and ergodicity for the stochastically forced Navier-Stokes equation, Commun. Math. Phys, vol.224, pp.83-106, 2001.

G. E. Falkovich, I. Kolokolov, V. Lebedev, and S. K. Turitsyn, Self-focusing in the perturbed and unperturbed nonlinear Schroedinger equation in critical dimension, J. Appl. Math, vol.60, pp.183-240, 2000.

F. Flandoli, Irreducibility of the 3-D Stochastic Navier???Stokes Equation, Journal of Functional Analysis, vol.149, issue.1, pp.160-177, 1997.
DOI : 10.1006/jfan.1996.3089

F. Flandoli and D. Gatarek, Martingale and stationnary solutions for stochastic Navier?Stokes equations, PTRF, pp.367-391, 1995.

F. Flandoli and B. Maslowski, Ergodicity of the 2-D Navier-Stokes equation under random perturbations, Communications in Mathematical Physics, vol.42, issue.1, pp.119-141, 1995.
DOI : 10.1007/BF02104513

F. Flandoli and M. Romito, Partial regularity for the stochastic Navier? Stokes equations, Transactions of the American Mathematical Society, vol.354, issue.06, pp.2207-2241, 2002.
DOI : 10.1090/S0002-9947-02-02975-6

F. Flandoli and M. Romito, Markov selections for the 3D stochastic Navier???Stokes equations, Probability Theory and Related Fields, vol.13, issue.1
DOI : 10.1007/s00440-007-0069-y

G. Furioli, P. G. Lemarié-rieusset, and E. Terraneo, Unicité dans L 3 (R 3 ) est d'autres espaces fonctionnel limites pour Navier?Stokes, Revisita Mathematica Iberoamericana, vol.16, issue.3, pp.605-667, 2000.

C. Foias and R. Temam, Gevrey class regularity for the solutions of the Navier-Stokes equations, Journal of Functional Analysis, vol.87, issue.2, pp.359-369, 1989.
DOI : 10.1016/0022-1236(89)90015-3

V. Ginzburg and L. Landau, On the theorie of superconductivity English transl, Zh. Eksp. Fiz. Physics : L.D. Landau, vol.20, pp.1064-546, 1950.

O. Goubet, Regularity of the attractor for a weakly damped nonlinear schr??dinger equation, Applicable Analysis, vol.58, issue.1-2, pp.99-119, 1996.
DOI : 10.1080/00036819608840420

M. Hairer, Exponential Mixing Properties of Stochastic PDEs Through Asymptotic Coupling, Proba. Theory Related Fields, pp.345-380, 2002.

M. Hairer and J. Mattingly, Ergodicity of the 2D Navier-Stokes equations with degenerate forcing, preprint

W. D. Henshaw, H. O. Kreiss, and L. G. Reyna, Smallest scale estimates for the Navier-Stokes equations for incompressible fluids, Archive for Rational Mechanics and Analysis, vol.9, issue.1, pp.21-44, 1990.
DOI : 10.1007/BF00431721

G. Huber and P. Alstrom, Universal decay of vortex density in two dimensions, Physica A: Statistical Mechanics and its Applications, vol.195, issue.3-4, pp.448-456, 1993.
DOI : 10.1016/0378-4371(93)90169-5

I. Karatzas and S. E. Shreve, Brownian motion and stochastic calculus, Graduate Texts in Mathematics, vol.113, 1991.

T. Kato, StrongL p -solutions of the Navier-Stokes equation inR m , with applications to weak solutions, Mathematische Zeitschrift, vol.74, issue.4, pp.471-480, 1984.
DOI : 10.1007/BF01174182

H. Koch and D. Tataru, Well posedeness for the Navier-Stokes equations, Advances in Math, pp.22-35, 2001.

S. Kuksin, On exponential convergence to a stationary mesure for nonlinear PDEs, The M. I. Viishik Moscow PDE seminar, Amer, Math. Soc. Trans, vol.206, issue.2, 2002.

S. Kuksin and A. Shirikyan, Stochastic Dissipative PDE's and Gibbs Measures, Communications in Mathematical Physics, vol.213, issue.2, pp.291-330, 2000.
DOI : 10.1007/s002200000237

S. Kuksin and A. Shirikyan, Ergodicity for the randomly forced 2D Navier-Stokes equations, Math. Phys. Anal. Geom, vol.4, 2001.

S. Kuksin and A. Shirikyan, A Coupling Approach??to Randomly Forced Nonlinear PDE's. I, Communications in Mathematical Physics, vol.221, issue.2, pp.351-366, 2001.
DOI : 10.1007/s002200100479

S. Kuksin, A. Piatnitski, and A. Shirikyan, A coupling approach to randomly forced randomly forced PDE's II, Communications in Mathematical Physics, vol.230, issue.1, pp.81-85, 2002.
DOI : 10.1007/s00220-002-0707-2

S. Kuksin and A. Shirikyan, Coupling approach to white-forced nonlinear PDEs, Journal de Math??matiques Pures et Appliqu??es, vol.81, issue.6, pp.567-602, 2002.
DOI : 10.1016/S0021-7824(02)01259-X

S. Kuksin and A. Shirikyan, Randomly forced CGL equation: stationary measures and the inviscid limit, Journal of Physics A: Mathematical and General, vol.37, issue.12, pp.3805-38222004
DOI : 10.1088/0305-4470/37/12/006

P. G. Lemarié-rieusset and E. Terraneo, Solutions faibles d'´ energie infinie pour leséquationsleséquations de Navier?Stokes dans R 3, C.R. Acad. Sci. Paris Sér. I Math, vol.238, issue.12, pp.1133-1138, 1999.

J. Leray, Sur le mouvement d'un liquide visqueux emplissant l'espace, Acta Mathematica, vol.63, issue.0, pp.193-248, 1934.
DOI : 10.1007/BF02547354

J. Mattingly, Exponential convergence for the stochastically forced Navier-Stokes equations and other partially dissipative dynamics, Communications in Mathematical Physics, vol.230, issue.3, pp.421-462, 2002.
DOI : 10.1007/s00220-002-0688-1

J. Mattingly and E. Pardoux, Ergodicity of the 2D Navier-Stokes Equations with Degenerate Stochastic Forcing, 2004.

R. Mikulevicius and B. Rozoskii, Stochastic Navier--Stokes Equations for Turbulent Flows, SIAM Journal on Mathematical Analysis, vol.35, issue.5, pp.1250-1310, 2004.
DOI : 10.1137/S0036141002409167

A. Newel and J. Whitehead, Finite bandwidth, finite amplitude convection, Journal of Fluid Mechanics, vol.39, issue.02, pp.279-303, 1969.
DOI : 10.1017/S0022112069000176

A. Newel and J. Whitehead, Review of the Finite Bandwidth Concept, Proceedings of the Internat, pp.284-289, 1971.
DOI : 10.1007/978-3-642-65073-4_39

C. Odasso, Ergodicity for the stochastic Complex Ginzburg?Landau equations , to appear in Annales de l'institut Henri-Poincaré, Probabilités et Statistiques

C. Odasso, Exponential mixing for stochastic PDEs: the non-additive case, Probability Theory and Related Fields, vol.6, issue.2
DOI : 10.1007/s00440-007-0057-2
URL : https://hal.archives-ouvertes.fr/hal-00004975

C. Odasso, Smoothness of the invariant measures of the 3D Navier? Stokes equations, preprint

C. Odasso, Exponential mixing for the 3D Navier-Stokes equation, pre- print

C. Odasso, Polynomial Mixing for weakly damped Stochastic Partial Differential Equations

A. Shirikyan, Analyticity of solutions of randomly perturbed twodimensional Navier-Stokes equations, Uspekhi Mat, Nauk translation in Russian Math. Surveys, vol.57, issue.57 4, pp.151-166, 2002.

A. Shirikyan, Exponential Mixing for 2D Navier-Stokes Equations Perturbed by an Unbounded Noise, Journal of Mathematical Fluid Mechanics, vol.6, issue.2, pp.169-193, 2004.
DOI : 10.1007/s00021-003-0088-0

R. Temam, Navier?Stokes Equations. Theory and Numerical Analysis., North-Holland, 1977.

R. Temam, Navier-Stokes equations and nonlinear functional analysis, Philadelphia (PA US) : SIAM , 1995 , CBMS-NSF regional conference series in applied mathematics

R. Temam, Infinite dimensional dynamical systems in mechanics and physics, Second edition , Applied mathematical sciences

M. Viot, Solutions faibles d'´ equations aux dérivées partielles stochastiques non-linéaires, 1976.

M. I. Vishik and A. Fursikov, Mathematical problem of statistical hydrodynamics, 1979.

W. Wahl, The equations of Navier?Stokes and abstract parabolic equations, Fried, 1985.

M. Barton-smith, Invariant measure for the stochastic Ginzburg Landau equation, NoDEA : Nonlinear Differential Equations and Applications, vol.11, issue.1, pp.29-52, 2004.
DOI : 10.1007/s00030-003-1040-y

P. Bebouche and A. , Inviscid Limits??of the Complex Ginzburg???Landau Equation, Communications in Mathematical Physics, vol.214, issue.1, pp.201-226, 2000.
DOI : 10.1007/s002200000263

J. Bricmont, A. Kupiainen, and R. Lefevere, Exponential mixing for the 2D stochastic Navier- Stokes dynamics, Communications in Mathematical Physics, vol.230, issue.1, pp.87-132, 2002.
DOI : 10.1007/s00220-002-0708-1

G. Da-prato and J. Zabczyk, Stochastic equations in infinite dimensions, Encyclopedia of Mathematics and its Applications, 1992.

A. De-bouard and A. Debussche, The stochastic non-linear Schrödinger equation in H 1 , Stochastic Analysis and applications 21, pp.197-126, 2003.

W. E. , J. C. Mattingly, and Y. G. Sinai, Gibbsian dynamics and ergodicity for the stochastically forced Navier-Stokes equation, Commun. Math. Phys, vol.224, pp.83-106, 2001.

V. Ginzburg and L. Landau, On the theorie of superconductivity English transl, Zh. Eksp. Fiz. Physics: L.D. Landau, vol.20, pp.1064-546, 1950.

M. Hairer, Exponential Mixing Properties of Stochastic PDEs Through Asymptotic Coupling, Proba. Theory Related Fields, pp.345-380, 2002.

S. Kuksin, On exponential convergence to a stationnary mesure for nonlinear PDEs, The M. I. Viishik Moscow PDE seminar, Amer, Math. Soc. Trans, vol.206, issue.2, 2002.

S. Kuksin and A. Shirikyan, Stochastic Dissipative PDE's and Gibbs Measures, Communications in Mathematical Physics, vol.213, issue.2, pp.291-330, 2000.
DOI : 10.1007/s002200000237

S. Kuksin and A. Shirikyan, A Coupling Approach??to Randomly Forced Nonlinear PDE's. I, Communications in Mathematical Physics, vol.221, issue.2, pp.351-366, 2001.
DOI : 10.1007/s002200100479

S. Kuksin and A. Shirikyan, Coupling approach to white-forced nonlinear PDEs, Journal de Math??matiques Pures et Appliqu??es, vol.81, issue.6, pp.567-602, 2002.
DOI : 10.1016/S0021-7824(02)01259-X

J. Mattingly, On recent progress for the stochastic Navier Stokes equations, Journées " ´ Equations aux Dérivées Partielles, 2003.
DOI : 10.5802/jedp.625

A. Newel and J. Whitehead, Finite bandwidth, finite amplitude convection, Journal of Fluid Mechanics, vol.39, issue.02, pp.279-303, 1969.
DOI : 10.1017/S0022112069000176

A. Newel and J. Whitehead, Review of the Finite Bandwidth Concept, Proceedings of the Internat, pp.284-289, 1971.
DOI : 10.1007/978-3-642-65073-4_39

C. Odasso, Propriétés ergodiques de l'´ equation de Ginzburg-Landau complexe bruitée, 2003.

P. Bebouche and A. , Inviscid Limits??of the Complex Ginzburg???Landau Equation, Communications in Mathematical Physics, vol.214, issue.1, pp.201-226, 2000.
DOI : 10.1007/s002200000263

J. Bricmont, A. Kupiainen, and R. Lefevere, Exponential mixing for the 2D stochastic Navier- Stokes dynamics, Communications in Mathematical Physics, vol.230, issue.1, pp.87-132, 2002.
DOI : 10.1007/s00220-002-0708-1

G. Da-prato and J. Zabczyk, Stochastic equations in infinite dimensions, Encyclopedia of Mathematics and its Applications, 1992.

G. Da-prato and J. Zabczyk, Ergodicity for Infinite Dimensional Systems, 1996.
DOI : 10.1017/CBO9780511662829

A. De-bouard and A. Debussche, The stochastic non-linear Schrödinger equation in H 1 , Stochastic Analysis and applications 21, pp.197-126, 2003.

W. E. , J. C. Mattingly, and Y. G. Sinai, Gibbsian dynamics and ergodicity for the stochastically forced Navier-Stokes equation, Commun. Math. Phys, vol.224, pp.83-106, 2001.

G. E. Falkovich, I. Kolokolov, V. Lebedev, and S. K. Turitsyn, Statistics of soliton-bearing systems with additive noise, Physical Review E, vol.63, issue.2, p.25601, 2001.
DOI : 10.1103/PhysRevE.63.025601

G. E. Falkovich, I. Kolokolov, V. Lebedev, and S. K. Turitsyn, Non-Gaussian error probability in optical soliton transmission, Physica D: Nonlinear Phenomena, vol.195, issue.1-2, pp.1-2, 2004.
DOI : 10.1016/j.physd.2004.01.044