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arXiv · 1902.00494

Mixing via controllability for randomly forced nonlinear dissipative PDEs

Sergei Kuksin·Vahagn Nersesyan·Armen Shirikyan
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Abstract

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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Sergei Kuksin, Vahagn Nersesyan, Armen Shirikyan. 2019-02-01. Mixing via controllability for randomly forced nonlinear dissipative PDEs. https://arxiv.org/abs/1902.00494

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