  Recently, a number of authors have investigated the conditions under which a
stochastic perturbation acting on an infinite dimensional dynamical system,
e.g. a partial differential equation, makes the system ergodic and mixing. In
particular, one is interested in finding minimal and physically natural
conditions on the nature of the stochastic perturbation. I shall review recent
results on this question; in particular, I shall discuss the Navier-Stokes
equation on a two dimensional torus with a random force which is white noise in
time, and excites only a finite number of modes. The number of excited modes
depends on the viscosity $\nu$, and grows like $\nu^{-3}$ when $\nu$ goes to
zero. This Markov process has a unique invariant measure and is exponentially
mixing in time.
