. .. , Invariant measures for PDEs: Motivations and approaches, p.91

. .. , Statement of the main result and comments, p.92

E. .. ,

. .. Kg, Almost conservation laws for

. .. Kg, , p.101

, The stochastic linear KG equation, exponential control

, Global dynamics of the damped-perturbed KG and well structuredness, p.103

. .. Kg, Step 2: Stationary measures and their estimations

E. .. Kg,

, Qualitative properties for the distribution of the Hamiltonian, p.111

, Global existence and uniqueness for the cubic KG equation in H 1,0 (D), p.115

L. Abdelouhab, J. L. Bona, M. Felland, and J. C. Saut, Nonlocal models for nonlinear, dispersive waves, Physica D: Nonlinear Phenomena, vol.40, issue.3, pp.360-392, 1989.

V. I. Arnold, Mathematical methods of classical mechanics, 1978.

J. Bourgain and A. Bulut, Invariant Gibbs measure evolution for the radial nonlinear wave equation on the 3d ball, J. Funct. Anal, vol.266, issue.4, pp.2319-2340, 2014.

T. B. Benjamin, Internal waves of permanent form in fluids of great depth, Journal of Fluid Mechanics, vol.29, issue.03, pp.559-592, 1967.

J. Bourgain, Periodic nonlinear Schrödinger equation and invariant measures, Comm. Math. Phys, vol.166, issue.1, pp.1-26, 1994.

N. Burq and F. Planchon, On well-posedness for the Benjamin-Ono equation, Math. Ann, vol.340, issue.3, pp.497-542, 2008.
URL : https://hal.archives-ouvertes.fr/hal-00017561

H. Brézis, Analyse fonctionnelle, Théorie et Applications. Masson, 1983.

N. Burq and N. Tzvetkov, Invariant measure for a three dimensional nonlinear wave equation, Art. ID rnm108, vol.26, 2007.

N. Burq and N. Tzvetkov, Random data Cauchy theory for supercritical wave equations. I. Local theory, Invent. Math, vol.173, issue.3, pp.449-475, 2008.
URL : https://hal.archives-ouvertes.fr/hal-00449551

N. Burq and N. Tzvetkov, Random data Cauchy theory for supercritical wave equations. II. A global existence result, Invent. Math, vol.173, issue.3, pp.477-496, 2008.
URL : https://hal.archives-ouvertes.fr/hal-00449551

N. Burq and N. Tzvetkov, Probabilistic well-posedness for the cubic wave equation, J. Eur. Math. Soc. (JEMS), vol.16, issue.1, pp.1-30, 2014.
URL : https://hal.archives-ouvertes.fr/hal-00575201

N. Burq, L. Thomann, and N. Tzvetkov, Remarks on the Gibbs measures for nonlinear dispersive equations, 2014.
URL : https://hal.archives-ouvertes.fr/hal-01097696

]. J. +-02, M. Colliander, G. Keel, H. Staffilani, T. Takaoka et al., Almost conservation laws and global rough solutions to a nonlinear Schrödinger equation, Math. Res. Lett, vol.9, issue.5-6, pp.659-682, 2002.

Y. Coudène, Théorie ergodique et systèmes dynamiques, 2013.

Y. Deng, Invariance of the Gibbs measure for the Benjamin-Ono equation, J. Eur. Math. Soc. (JEMS), 2015.

G. Da-prato and J. Zabczyk, Stochastic equations in infinite dimensions, 2014.

A. De-suzzoni, Invariant measure for the cubic wave equation on the unit ball of R 3, Dyn. Partial Differ. Equ, vol.8, issue.2, pp.127-147, 2011.

A. De-suzzoni, Invariant measure for the Klein-Gordon equation in a non periodic setting, 2014.

Y. Deng, N. Tzvetkov, and N. Visciglia, Invariant measures and long time behaviour for the Benjamin-Ono equation III, Comm. Math. Phys, vol.339, issue.3, pp.815-857, 2015.

A. Einstein, Investigations on the theory of the Brownian movement

P. M. Edwin and B. Roberts, The benjamin-ono-burgers equation: an application in solar physics, Wave Motion, vol.8, issue.2, pp.151-158, 1986.

M. I. Freidlin and A. D. Wentzell, Random perturbations, Random Perturbations of Dynamical Systems, pp.15-43, 1984.

I. Gyöngy and N. V. Krylov, On stochastics equations with respect to semimartingales. II. Itô formula in Banach spaces, Stochastics, vol.6, issue.3-4, p.82, 1981.

A. Ionescu and C. Kenig, Global well-posedness of the Benjamin-Ono equation in low-regularity spaces, Journal of the American Mathematical Society, vol.20, issue.3, pp.753-798, 2007.

A. Ionescu and C. Kenig, Local and global well-posedness of periodic KP-I equations. Mathematical Aspects of Nonlinear Dispersive Equations, Ann. Math. Stud, vol.163, pp.181-211, 2009.

F. Jedrzejewski, Modeles aleatoires et physique probabiliste, 2009.

F. W. King, Hilbert transforms, 2009.

B. Koopman, Hamiltonian systems and transformation in hilbert space, Proceedings of the National Academy of Sciences, vol.17, issue.5, pp.315-318, 1931.

T. Kappeler, J. Pöschel, &. Kdv, and . Kam, , vol.45, 2013.

U. Krengel, Ergodic theorems, vol.59, 1985.

N. V. Krylov, On estimates of the maximum of a solution of a parabolic equation and estimates of the distribution of a semimartingale, Mathematics of the USSR-Sbornik, vol.58, issue.1, p.207, 1987.

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

S. Kuksin and A. Shirikyan, Randomly forced CGL equation: stationary measures and the inviscid limit, Journal of Physics. A. Mathematical and General, vol.37, pp.3805-3822, 2004.

S. Kuksin and A. Shirikyan, Mathematics of Two-Dimensional Turbulence, 2012.

H. Koch and N. Tzvetkov, On the local well-posedness of the Benjamin-Ono equation in H s (R), International Mathematics Research Notices, issue.26, pp.1449-1464, 2003.

S. Kuksin, The Eulerian limit for 2D statistical hydrodynamics, J. Statist. Phys, vol.115, issue.1-2, pp.469-492, 2004.

S. Kuksin, On distribution of energy and vorticity for solutions of 2d Navier-Stokes equation with small viscosity, Communications in Mathematical Physics, vol.284, issue.2, pp.407-424, 2008.
URL : https://hal.archives-ouvertes.fr/hal-00832704

J. L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires. Dunod, 1969.

J. L. Lebowitz, H. A. Rose, and E. R. Speer, Statistical mechanics of the nonlinear Schrödinger equation, J. Statist. Phys, vol.50, issue.3-4, pp.657-687, 1988.

D. Martirosyan, Exponential mixing for the white-forced damped nonlinear wave equation, Evol. Equ. Control Theory, vol.3, issue.4, pp.645-670, 2014.
URL : https://hal.archives-ouvertes.fr/hal-00980358

Y. Matsuno, Bilinear transformation method, Mathematics in Science and Engineering, 1984.

L. Molinet, Global well-posedness in L 2 for the periodic Benjamin-Ono equation, American Journal of Mathematics, vol.130, issue.3, pp.635-683, 2008.
URL : https://hal.archives-ouvertes.fr/hal-00016752

L. Molinet and D. Pilod, The Cauchy problem for the Benjamin-Ono equation in L 2 revisited, Analysis & PDE, vol.5, issue.2, pp.365-395, 2012.
URL : https://hal.archives-ouvertes.fr/hal-00499346

L. Molinet, J. C. Saut, and N. Tzvetkov, Ill-posedness issues for the Benjamin-Ono and related equations, SIAM journal on mathematical analysis, vol.33, issue.4, pp.982-988, 2001.

A. Nakamura, Bäcklund transform and conservation laws of the Benjamin-Ono equation, J. Phys. Soc. Japan, vol.47, issue.4, pp.1335-1340, 1979.

A. R. Nahmod, T. Oh, L. Rey-bellet, and G. Staffilani, Invariant weighted Wiener measures and almost sure global well-posedness for the periodic derivative NLS, J. Eur. Math. Soc, vol.14, issue.4, pp.1275-1330, 2012.

T. Oh, Invariance of the Gibbs measure for the Schrödinger-Benjamin-Ono system, SIAM J. Math. Anal, vol.41, issue.6, pp.2207-2225, 2009.

B. , Stochastic differential equations

H. Ono, Algebraic solitary waves in stratified fluids, Journal of the Physical Society of Japan, vol.39, issue.4, pp.1082-1091, 1975.

T. Oh and O. Pocovnicu, Probabilistic global well-posedness of the energy-critical defocusing quintic nonlinear wave equation on R 3, J. Math. Pures Appl, vol.105, issue.9, pp.342-366, 2016.

T. Oh and N. Tzvetkov, Quasi-invariant Gaussian measures for the cubic fourth order nonlinear schrödinger equation, Probability Theory and Related Fields, pp.1-48, 2015.

E. Pardoux, Equations aux dérivées partielles stochastiques non linéaires monotones. Etude de solutions fortes de type Ito, 1975.

J. Perrin, Mouvement brownien et réalité moléculaire, Annales de Chimie et de Physique, vol.18, pp.5-104, 1909.

H. Poincaré, Sur le problème des trois corps et les équations de la dynamique, Acta mathematica, vol.13, issue.1, 1890.

B. Roberts and . Mangeney, Solitons in solar magnetic flux tubes, Monthly Notices of the Royal Astronomical Society, vol.198, issue.1, pp.7-11, 1982.

A. Shirikyan, Qualitative properties of stationary measures for three-dimensional Navier-Stokes equations, J. Funct. Anal, vol.249, issue.2, pp.284-306, 2007.

A. Shirikyan, Local times for solutions of the complex Ginzburg-Landau equation and the inviscid limit, J. Math. Anal. Appl, vol.384, issue.1, pp.130-137, 2011.
URL : https://hal.archives-ouvertes.fr/hal-00461510

M. Sy, Invariant measure and long time behavior of regular solutions of the BenjaminOno equation, 2016.

M. Sy, Invariant measure and large time dynamics for the Klein-Gordon equation in 3D, 2017.

T. Tao, Global well-posedness of the Benjamin-Ono equation in H 1 (R), Journal of Hyperbolic Differential Equations, vol.1, issue.01, pp.27-49, 2004.

T. Tao, Nonlinear dispersive equations: local and global analysis, vol.106, 2006.

T. Tao, Lectures on ergodic theory, 2008.

L. Thomann, Invariant Gibbs measures for dispersives PDEs, 2016.

L. Thomann and N. Tzvetkov, Gibbs measure for the periodic derivative nonlinear Schrödinger equation, Nonlinearity, vol.23, issue.11, pp.2771-2791, 2010.

N. Tzvetkov and N. Visciglia, Gaussian measures associated to the higher order conservation laws of the Benjamin-Ono equation, Ann. Sci. Éc. Norm. Supér, vol.46, issue.4, pp.249-299, 2013.

N. Tzvetkov and N. Visciglia, Invariant measures and long-time behavior for the Benjamin-Ono equation, Int. Math. Res. Not. IMRN, issue.17, pp.4679-4714, 2014.

N. Tzvetkov and N. Visciglia, Invariant measures and long time behaviour for the Benjamin-Ono equation II, J. Math. Pures Appl, vol.103, issue.9, pp.102-141, 2015.

N. Tzvetkov, Ill-posedness issues for nonlinear dispersive equations, Lectures on nonlinear dispersive equations, vol.27, pp.63-103, 2006.
URL : https://hal.archives-ouvertes.fr/hal-00108211

N. Tzvetkov, Construction of a Gibbs measure associated to the periodic Benjamin-Ono equation. Probability theory and related fields, vol.146, pp.481-514, 2010.

N. Tzvetkov, Quasiinvariant Gaussian measures for one-dimensional Hamiltonian partial differential equations, Forum Math. Sigma, vol.3, p.28, 2015.

N. Tzvetkov, On Hamiltonian partial differential equations with random initial conditions, Lecture notes, 2016.

M. Willem, Analyse fonctionnelle élémentaire. Cassini, 2003.

S. Xu, Invariant Gibbs measure for 3D NLW in infinite volume, 2014.

P. E. Zhidkov, Korteweg-de Vries and nonlinear Schrödinger equations: qualitative theory. Number 1756, 2001.

P. E. Zhidkov, On an infinite sequence of invariant measures for the cubic nonlinear Schrödinger equation, Int. J. Math. Math. Sci, vol.28, issue.7, pp.375-394, 2001.