We discuss a proof of uniqueness of invariant measures for the damped-driven stochastic Korteweg-de Vries equation, where the noise is additive and sufficiently non-degenerate. It is shown that a simple, but versatile control strategy, typically employed to establish exponential mixing for strongly dissipative systems, can nevertheless be applied in this weakly dissipative setting to obtain unique ergodicity, albeit without mixing rates. Under the assumption of large damping, however, exponential mixing can be recovered and, as a byproduct, the regularity of the support of the invariant measure is deduced as well. Time permitting, application of this approach to other weakly dissipative systems, such as the damped-driven Nonlinear Schrodinger equation will also be discussed.

This is joint work with Nathan Glatt-Holtz and Geordie Richards.