Unique Ergodicity and Mixing for the Degenerately forced Boussinesq Equations and related systems

Unique Ergodicity and Mixing for the Degenerately forced Boussinesq Equations and related systems

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Nathan Glatt-Holtz , Virginia Tech
Fine Hall 322

We establish the existence, uniqueness and attraction properties of an ergodic invariant measure for the Boussinesq Equations in the presence of a degenerate stochastic forcing acting only in the temperature equation and only at the largest spatial scales. The central challenge is to establish time asymptotic smoothing properties of the Markovian dynamics corresponding to this system. Towards this aim we encounter a Lie bracket structure in the associated vector fields with a complicated dependence on solutions. This leads us to develop a novel Hormander-type condition for infinite-dimensional systems. Demonstrating the sufficiency of this condition requires new techniques for the spectral analysis of the Malliavin covariance matrix. This is joint work with Juraj Foldes, Geordie Richards and Enrique Thomann.