  We study the motion of a particle in a random time-dependent vector field
defined by the 2D Navier-Stokes system with a noise. Under suitable
non-degeneracy hypotheses we prove that the empirical measures of the
trajectories of the pair (velocity field, particle) satisfy the LDP with a good
rate function. Moreover, we show that the law of a unique stationary solution
restricted to the particle component possesses a positive smooth density with
respect to the Lebesgue measure in any finite time. This allows one to define a
natural concept of the entropy production, and to show that its time average is
a bounded function of the trajectory. The proofs are based on a new criterion
for the validity of the level-3 LDP for Markov processes and an application of
a general result on the image of probability measures under smooth maps to the
laws associated with the motion of the particle.
