| Abstract: |
| We present a general set of conditions on a Markov process which guarantee that the corresponding Markov operator satisfies the Asymptotic Strong Feller property. The main examples for which these conditions are satisfied include the damped-driven Korteweg-de Vries (KdV) and damped-driven cubic nonlinear Schrodinger (NLS) equations both forced with sufficiently non-degenerate noise. These conditions generalize those in current frameworks which typically include equations with stronger dissipative mechanisms such as the 2D Navier-Stokes equations, 2D Boussinesq equations, or strongly damped Euler equations, but not the damped-driven KdV or NLS equations. As an application, we additionally discuss a proof of the controllability of the damped KdV equation, which thus allows one to deduce the uniqueness of the invariant measure for its stochastically forced counterpart. |
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