The evolution of measures determined by the Navier–Stokes equations and on the solvability of the Cauchy problem for the Hopf statistical equation

We prove the solvability of the Cauchy problem for Hopf's statistical equation, corresponding to the general three-dimensional initial- and boundary-value problem for the Navier–Stokes equations, with the assumption that the exterior forces and the boundary conditions are fixed while the initial field of the velocities is stochastic. Preliminarily we construct a family of measurable single-valued mappings $W_t$ defining the evolution $\mu_t$ of the probability measure $\mu$, given on the metric space $Y_R$ of the initial field of velocities according to the formula: $\mu_t(\omega)=\mu(W_t^{-1}\omega)$, where $\omega$ is any set from the $\sigma$-algebra $\Sigma(Y_R)$ of analytic sets of the space $Y_R$.