Uniqueness of Solutions of the Stochastic Navier–stokes Equation with Invariant Measure given by the Enstrophy

A stochastic Navier–Stokes equation with space-time Gaussian white noise is considered, having as infinitesimal invariant measure a Gaussian measure µν whose covariance is given in terms of the enstrophy. Pathwise uniqueness for µν-a.e. initial velocity is proven for solutions having µν as invariant measure. 1. Introduction. We are interested in the stochastic Navier–Stokes equation with a space-time white noise. We consider the spatial domain to be the torus T 2 = [0, 2π] 2 (hence periodic boundary conditions are assumed). In [1] it has been shown that there exists an infinitesimal invariant measure associated to this stochastic equation; this is a Gaussian measure µ ν , with covariance given in terms of the enstrophy (and of the viscosity parameter ν). Existence of a solution has been proven in two different ways: [1] considers a weak solution and [9] a strong solution (weak and strong are to be understood in the probabilistic sense). The common point of these papers is that the solution is obtained as the limit of Galerkin approximations. No result of uniqueness has been given in [1], whereas [9] shows existence and uniqueness in a smaller class than the natural one to consider for this problem. Indeed, the statement of Theorem 5.1 in [9] involves an auxiliary process (denoted by z in Section 4), not appearing in the given stochastic Navier– Stokes equation, and for this reason the definition of uniqueness given in