Exponential mixing for stochastic PDEs: the non-additive case

We establish a general criterion which ensures exponential mixing of parabolic stochastic partial differential equations (SPDE) driven by a non additive noise which is white in time and smooth in space. We apply this criterion on two representative examples: 2D Navier–Stokes (NS) equations and Complex Ginzburg–Landau (CGL) equation with a locally Lipschitz noise. Due to the possible degeneracy of the noise, Doob theorem cannot be applied. Hence, a coupling method is used in the spirit of Kuksin and Shirikyan (J. Math. Pures Appl. 1:567–602, 2002) and Mattingly (Commun. Math. Phys. 230:421–462, 2002). Previous results require assumptions on the covariance of the noise which might seem restrictive and artificial. For instance, for NS and CGL, the covariance operator is supposed to be diagonal in the eigenbasis of the Laplacian and not depending on the high modes of the solutions. The method developed in the present paper gets rid of such assumptions and only requires that the range of the covariance operator contains the low modes.