Comptes Rendus
no. 6
Partial differential equations
Existence of invariant measures for some damped stochastic dispersive equations
[Existence de mesures invariantes pour des équations dispersives stochastiques]

Présenté par : the Editorial Board

Ibrahim Ekren1 ; Igor Kukavica1 ; Mohammed Ziane2
1 Departement fur Mathematik, ETH Zurich, Ramistrasse 101, CH-8092, Zurich, Switzerland
2 Department of Mathematics, University of Southern California, Los Angeles, CA 90089-2532, USA
Comptes Rendus. Mathématique, Volume 355 (2017) no. 6, pp. 676-679.

On étudie le comportement asymptotique des solutions d'équations dispersives stochastiques amorties, en particulier les équation de KdV et les équations de Schrödinger. Nous montrons que le semi-groupe de transition est Feller et nous établissons l'existence d'une mesure invariante grâce à la propriété de compacité asymptotique du semi-groupe de transition et au critère d'Aldous.

We address the long-time behavior of solutions to damped dispersive stochastic partial differential equations, namely the KdV equation and the nonlinear Schrödinger equation on the whole space. We prove that the transition semigroup is Feller and establish the existence of an invariant measure using the asymptotic compactness property of the transition semigroup and the Aldous criterion.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2017.04.018

Ibrahim Ekren 1 ; Igor Kukavica 1 ; Mohammed Ziane 2

1 Departement fur Mathematik, ETH Zurich, Ramistrasse 101, CH-8092, Zurich, Switzerland
2 Department of Mathematics, University of Southern California, Los Angeles, CA 90089-2532, USA 
@article{CRMATH_2017__355_6_676_0,
     author = {Ibrahim Ekren and Igor Kukavica and Mohammed Ziane},
     title = {Existence of invariant measures for some damped stochastic dispersive equations},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {676--679},
     publisher = {Elsevier},
     volume = {355},
     number = {6},
     year = {2017},
     doi = {10.1016/j.crma.2017.04.018},
     language = {en},
}
TY  - JOUR
AU  - Ibrahim Ekren
AU  - Igor Kukavica
AU  - Mohammed Ziane
TI  - Existence of invariant measures for some damped stochastic dispersive equations
JO  - Comptes Rendus. Mathématique
PY  - 2017
SP  - 676
EP  - 679
VL  - 355
IS  - 6
PB  - Elsevier
DO  - 10.1016/j.crma.2017.04.018
LA  - en
ID  - CRMATH_2017__355_6_676_0
ER  - 
%0 Journal Article
%A Ibrahim Ekren
%A Igor Kukavica
%A Mohammed Ziane
%T Existence of invariant measures for some damped stochastic dispersive equations
%J Comptes Rendus. Mathématique
%D 2017
%P 676-679
%V 355
%N 6
%I Elsevier
%R 10.1016/j.crma.2017.04.018
%G en
%F CRMATH_2017__355_6_676_0
Ibrahim Ekren; Igor Kukavica; Mohammed Ziane. Existence of invariant measures for some damped stochastic dispersive equations. Comptes Rendus. Mathématique, Volume 355 (2017) no. 6, pp. 676-679. doi : 10.1016/j.crma.2017.04.018. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2017.04.018/

[1] P. Constantin; N. Glatt-Holtz; V. Vicol Unique ergodicity for fractionally dissipated, stochastically forced 2D Euler equations, Commun. Math. Phys., Volume 330 (2014) no. 2, pp. 819-857

[2] G. Da Prato; J. Zabczyk Ergodicity for Infinite-Dimensional Systems, London Mathematical Society Lecture Note Series, vol. 229, Cambridge University Press, Cambridge, UK, 1996

[3] A. de Bouard; A. Debussche On the stochastic Korteweg–de Vries equation, J. Funct. Anal., Volume 154 (1998) no. 1, pp. 215-251

[4] A. de Bouard; A. Debussche The stochastic nonlinear Schrödinger equation in H1, Stoch. Anal. Appl., Volume 21 (2003) no. 1, pp. 97-126

[5] A. Debussche; C. Odasso Ergodicity for a weakly damped stochastic non-linear Schrödinger equation, J. Evol. Equ., Volume 5 (2005) no. 3, pp. 317-356

[6] I. Ekren; I. Kukavica; M. Ziane Existence of invariant measures for the stochastic damped KdV equation, Indiana Univ. Math. J. (2017) (in press)

[7] I. Ekren; I. Kukavica; M. Ziane Existence of invariant measures for the stochastic damped Schrödinger equation, Stoch. Partial Differ. Equ., Anal. Computat. (2017) (in press)

[8] F. Flandoli Dissipativity and invariant measures for stochastic Navier–Stokes equations, Nonlinear Differ. Equ. Appl., Volume 1 (1994) no. 4, pp. 403-423

[9] O. Goubet Regularity of the attractor for a weakly damped nonlinear Schrödinger equation, Appl. Anal., Volume 60 (1996) no. 1–2, pp. 99-119

[10] N. Glatt-Holtz; J. Mattingly; G. Richards On unique ergodicity in nonlinear stochastic partial differential equations, J. Stat. Phys. (2017) (in press)

[11] M. Hairer; J.C. Mattingly Ergodicity of the 2D Navier–Stokes equations with degenerate stochastic forcing, Ann. of Math. (2), Volume 164 (2006) no. 3, pp. 993-1032

[12] J.U. Kim Invariant measures for a stochastic nonlinear Schrödinger equation, Indiana Univ. Math. J., Volume 55 (2006) no. 2, pp. 687-717

[13] Y. Prokhorov Convergence of random processes and limit theorems in probability theory, Theory Probab. Appl., Volume 2 (1956) no. 2, pp. 157-214

[14] R. Rosa The global attractor of a weakly damped, forced Korteweg–de Vries equation in H1(R), Rio de Janeiro, 1999 (Mat. Contemp.), Volume vol. 19 (2000), pp. 129-152

[15] R. Temam Sur un problème non linéaire, J. Math. Pures Appl. (9), Volume 48 (1969), pp. 159-172

Cité par Sources :

Commentaires - Politique