[Mini-course] Ergodicity via controllability, IV

The problem of ergodicity of randomly forced dissipative PDEs has attracted a lot of attention in the last twenty years. It is well understood that if all or sufficiently many Fourier modes of the PDE are directly perturbed by the noise, then the problem has a unique stationary measure which is exponentially stable in an appropriate metric. The case when the random perturbation acts directly only on few Fourier modes is much less understood and is the main subject of this minicourse. 

We will explain how the controllability properties of the underlying deterministic system can be used to study the ergodic behavior of the stochastic dynamics. We will carefully introduce all the key concepts coming both from control theory and probability. The results will be illustrated through the examples of 2D Navier-Stokes and Ginzburg-Landau equations; however, the methods apply to a wide variety of systems as soon as they satisfy appropriate controllability conditions. 

This course is based on recent joint works with Sergei Kuksin and Armen Shirikyan.

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This mini-course will run weekly from 3:30pm to 5:30pm on Wednesdays.