arXiv:1106.3861v2 [math.AP] 6 Sep 2011 Solution to the Navier-Stokes equations with random initial data Evelina Shamarova Abstract We construct a solution to the spatially periodic d-dimensional Navier–Stokes equations with a given distribution of the intial data. The solution takes values in the Sobolev space H α , where the in- dex α ∈ R is fixed arbitrary. The distribution of the initial value is a Gaussian measure on H α whose parameters depend on α. The Navier– Stokes solution is then a stochastic process verifying the Navier–Stokes equations almost surely. It is obtained as a limit in distribution of solutions to finite-dimensional ODEs which are Galerkin-type approx- imations for the Navier–Stokes equations. Moreover, the constructed Navier–Stokes solution U (t, ω) possesses the property: E  f (U (t, ω))  =  H α f (e tνΔ u)γ (du), where f ∈ L 1 (γ ), e tΔ is the heat semigroup, ν is the viscosity in the Navier–Stokes equations, and γ is the distribution of the initial data. 1. Introduction Among the abundant amount of literature on the Navier–Stokes equations just a relatively smaller number of works treat this problem from the infi- nite dimensional analysis point of view, i.e. by reducing the Navier–Stokes equations to an infinite-dimensional problem. We mention here the papers [1, 2, 3, 9, 10, 12] that use infinite-dimensional approaches. In the present paper we prove the existence of a Sobolev space valued solu- tion to the Navier–Stokes equations on the torus T d with the given Gaussian distribution of the initial data. The result of [1] related to the Euler equation on T 2 follows from our result as a particular case (d = 2, ν = 0). Moreover, 1