Mixing via controllability for randomly forced nonlinear dissipative PDEs
[Mélange via la contrôlabilité pour des EDP non linéaires dissipatives bruitées]
Journal de l'École polytechnique — Mathématiques, Tome 7 (2020) , pp. 871-896.

Dans l’article [KNS20], nous avons étudié le problème de mélange pour une classe d’EDP avec un bruit borné très dégénéré et établi l’unicité de la mesure stationnaire et sa stabilité exponentielle pour la métrique dual-Lipschitz. L’une des hypothèses imposées au problème en question exigeait que l’équation non perturbée ait exactement un point d’équilibre globalement stable. Dans cet article, on assouplit cette hypothèse, en ne supposant que la contrôlabilité globale à un point donné. On prouve que l’unicité d’une mesure stationnaire et la convergence restent vraies, alors que le taux de convergence n’est pas nécessairement exponentiel. Le résultat est applicable aux EDP de type parabolique avec une perturbation aléatoire, à condition que la partie déterministe de la force extérieure soit en position générale, ce qui garantit une structure régulière pour l’attracteur du problème non perturbé.

In the paper [KNS20], we studied the problem of mixing for a class of PDEs with a very degenerate bounded noise and established the uniqueness of stationary measure and its exponential stability in the dual-Lipschitz metric. One of the hypotheses imposed on the problem in question required that the unperturbed equation should have exactly one globally stable equilibrium point. In this paper, we relax that condition, assuming only global controllability to a given point. It is proved that the uniqueness of a stationary measure and convergence to it are still valid, whereas the rate of convergence is not necessarily exponential. The result is applicable to randomly forced parabolic-type PDEs, provided that the deterministic part of the external force is in general position, ensuring a regular structure for the attractor of the unperturbed problem.

Reçu le : 2019-03-18
Accepté le : 2020-04-29
Publié le : 2020-05-27
DOI : https://doi.org/10.5802/jep.130
Classification : 35K58,  35R60,  37A25,  37L55,  60G50,  60H15,  76M35,  93B18,  93C20
Mots clés: Processus markoviens, mesure stationnaire, mélange, EDP paraboliques non linéaires, fonction de Lyapunov, série de Haar
@article{JEP_2020__7__871_0,
     author = {Sergei Kuksin and Vahagn Nersesyan and Armen Shirikyan},
     title = {Mixing via controllability for randomly~forced nonlinear dissipative PDEs},
     journal = {Journal de l'\'Ecole polytechnique --- Math\'ematiques},
     publisher = {\'Ecole polytechnique},
     volume = {7},
     year = {2020},
     pages = {871-896},
     doi = {10.5802/jep.130},
     language = {en},
     url = {jep.centre-mersenne.org/item/JEP_2020__7__871_0/}
}
Sergei Kuksin; Vahagn Nersesyan; Armen Shirikyan. Mixing via controllability for randomly forced nonlinear dissipative PDEs. Journal de l'École polytechnique — Mathématiques, Tome 7 (2020) , pp. 871-896. doi : 10.5802/jep.130. https://jep.centre-mersenne.org/item/JEP_2020__7__871_0/

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