The Kolmogorov equation for a 2D-Navier–Stokes stochastic flow in a channel

We consider a 2D-Navier–Stokes equation in a channel with periodic conditions along the axis, Navier type conditions on the wall and perturbed by a stochastic driving force QẆ where QQ is a nonnegative, self-adjoint operator of trace class and Ẇ is a space-time white noise. This work is concerned with the construction of the Kolmogorov operator associated with the corresponding stochastic process expressed in terms of vorticity. The main result is that the Kolmogorov operator, defined on a space of smooth C2C2-functions, is essentially mm-dissipative in L2(H0,μ)L2(H0,μ) where H0H0 is a state space and μμ an invariant measure.