In this chapter, we study limiting theorems for the 2D Navier-Stokes system with random perturbations. To simplify the presentation, we shall confine ourselves to the case of spatially regular white noise; however, all the results remain true for random kick forces. The first section is devoted to the derivation of the strong law of large numbers (SLLN), the law of the iterated logarithm (LIL), and the central limit theorem (CLT). Our approach is based on the reduction of the problem to similar questions for martingales and an application of some general results on SLLN, LIL, and CLT. In Section 4.2, we study the relationship between stationary distributions and random attractors. Roughly speaking, it is proved that the support of the random probability measure obtained by the disintegration of the unique stationary distribution is a random point attractor for the RDS in question. The third section deals with the stationary distributions for the Navier-Stokes system perturbed by a random force depending on a parameter. We first prove that the stationary measures continuously depend on spatially regular white noise. We next consider high-frequency random kicks and show that, under suitable normalisation, the corresponding family of stationary measures converges weakly to the unique stationary distribution corresponding to the white-noise perturbation. Finally, in Section 4.4, we discuss the physical relevance of the results of this chapter.

Originally, the main body of these exercises was developed for, and presented to, the students in the Magistère des Universités Parisiennes between 1984 and 1990; the audience consisted mainly of students from the Écoles Normales, and the spirit of the Magistère was to blend “undergraduate probability” (≟ random variables, their distributions, and so on …) with a first approach to “graduate probability” (≟ random processes). Later, we also used these exercises, and added some more, either in the Préparation à l'Agrégation de Mathématiques, or in more standard Master courses in probability.

In order to fit the exercises (related to the lectures) in with the two levels alluded to above, we systematically tried to strip a number of results (which had recently been published in research journals) of their random processes apparatus, and to exhibit, in the form of exercises, their random variables skeleton.

Of course, this kind of reduction may be done in almost every branch of mathematics, but it seems to be a quite natural activity in probability theory, where a random phenomenon may be either studied on its own (in a “small” probability world), or as a part of a more complete phenomenon (taking place in a “big” probability world); to give an example, the classical central limit theorem, in which only one Gaussian variable (or distribution) occurs in the limit, appears, in a number of studies, as a one-dimensional “projection” of a central limit theorem involving processes, in which the limits may be several Brownian motions, the former Gaussian variable appearing now as the value at time 1, say, of one of these Brownian motions.