Mixing via controllability for randomly forced nonlinear dissipative PDEs
Journal de l’École polytechnique — Mathématiques, Volume 7 (2020), pp. 871-896.

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.

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.

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Accepted:
Published online:
DOI: 10.5802/jep.130
Classification: 35K58,  35R60,  37A25,  37L55,  60G50,  60H15,  76M35,  93B18,  93C20
Keywords: Markov process, stationary measure, mixing, nonlinear parabolic PDEs, Lyapunov function, Haar series
Sergei Kuksin 1; Vahagn Nersesyan 2; Armen Shirikyan 3

1 Institut de Mathématiques de Jussieu–Paris Rive Gauche, CNRS, Université Paris Diderot, UMR 7586, Sorbonne Paris Cité F-75013, Paris, France & School of Mathematics, Shandong University, Jinan, PRC & Saint Petersburg State University, Universitetskaya nab., St. Petersburg, Russia
2 Laboratoire de Mathématiques, UMR CNRS 8100, UVSQ, Université Paris-Saclay 45, av. des Etats-Unis, F-78035 Versailles, France & Centre de Recherches Mathématiques, CNRS UMI 3457, Université de Montréal Montréal, QC, H3C 3J7, Canada
3 Department of Mathematics, CY Cergy Paris Université, CNRS UMR 8088 2 avenue Adolphe Chauvin, 95302 Cergy–Pontoise, France & Department of Mathematics and Statistics, McGill University 805 Sherbrooke Street West, Montreal, QC, H3A 2K6, Canada
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Sergei Kuksin; Vahagn Nersesyan; Armen Shirikyan. Mixing via controllability for randomly forced nonlinear dissipative PDEs. Journal de l’École polytechnique — Mathématiques, Volume 7 (2020), pp. 871-896. doi : 10.5802/jep.130. https://jep.centre-mersenne.org/articles/10.5802/jep.130/

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