Based on a previously introduced downscaling data assimilation algorithm,
which employs a nudging term to synchronize the coarse mesh spatial scales, we
construct a determining map for recovering the full trajectories from their
corresponding coarse mesh spatial trajectories, and investigate its properties.
This map is then used to develop a downscaling data assimilation scheme for
statistical solutions of the two-dimensional Navier-Stokes equations, where the
coarse mesh spatial statistics of the system is obtained from discrete spatial
measurements. As a corollary, we deduce that statistical solutions for the
Navier-Stokes equations are determined by their coarse mesh spatial
distributions. Notably, we present our results in the context of the
Navier-Stokes equations; however, the tools are general enough to be
implemented for other dissipative evolution equations
