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Article Dans Une Revue Journal de l'École polytechnique — Mathématiques Année : 2020

Mixing via controllability for randomly forced nonlinear dissipative PDEs

Résumé

We continue our study of the problem of mixing for a class of PDEs with very degenerate noise. As we established earlier, the uniqueness of stationary measure and its exponential stability in the dual-Lipschitz metric holds under the hypothesis that the unperturbed equation has 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. The proof uses a new idea that reduces the verification of a stability property to the investigation of a conditional random walk.
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Dates et versions

hal-02004635 , version 1 (01-02-2019)

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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, 2020, 7, pp.871-896. ⟨10.5802/jep.130⟩. ⟨hal-02004635⟩
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