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.

我们研究由二维Navier–Stokes系统定义的带有噪声的随机时间相关矢量场中粒子的运动。在适当的非简并性假设下，我们证明了该对轨迹（速度场，粒子）的经验测度以良好的速率函数满足LDP。此外，我们证明了在任何有限时间内，相对于Lebesgue测度，受限于粒子成分的唯一平稳解的定律具有正的平滑密度。这使人们可以定义熵产生的自然概念，并表明其时间平均值是轨迹的有界函数。