2002/12/01 by Jean Bricmont
Mathematics · #math.PR #msc:35Q30 #msc:60H15
published as Proceedings of the ICM, Beijing 2002, vol. 1, 567--585
arxiv created 2002/12/01 · arxiv updated 2009/12/01
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 ν, and grows like ν-3 when ν goes to zero. This Markov process has a unique invariant measure and is exponentially mixing in time.