Unique Measure for Time-Dependent Random PDEs

Abstract This paper proves the uniqueness of measure for time-inhomogeneous random dynamical systems that exhibit particular controllability and coupling conditions. This is done by extending a result for uniqueness of measure for timehomogeneous Markov processes to the time-inhomogeneous case, showing that the Markov process is exponentially mixing in the dual-Lipschitz norm. It is then shown that the 2D Navier-Stokes equations on the sphere with a time-dependent deterministic force and a ”kick”-type random perturbation satisfy the conditions and thus have a unique limiting measure. MSC Classification: 35Q30 , 60H15 , 60J05 , 93C20 , 35R01 , 60J99