Existence and uniqueness of stationary solutions for 3D Navier-Stokes system with small random Forcing via stochastic cascades

We consider the 3D Navier-Stokes system in the Fourier space with regular forcing given by a stationary in time stochastic process satisfying a smallness condition. We explicitly construct a stationary solution of the system and prove a uniqueness theorem for this solution in the class of functions with Fourier transform majorized by a certain function h. Moreover we prove the following "one force-one solution" principle: the unique stationary solution at time t is presented as a functional of the realization of the forcing in the past up to t. Our explicit construction of the solution is based upon the stochastic cascade representation.