## Abstract

We study a class of dissipative nonlinear PDE's forced by a random force ?^{?}(t, x), with the space variable x varying in a bounded domain. The class contains the 2D Navier-Stokes equations (under periodic or Dirichlet boundary conditions), and the forces we consider are those common in statistical hydrodynamics: they are random fields smooth in x and stationary, short-correlated in time t. In this paper, we confine ourselves to "kick forces" of the form ?^{?}(t, x) = S^{+8}_{k=-8} d(t - kT)?_{k}(x), where the ?_{k}'s are smooth bounded identically distributed random fields. The equation in question defines a Markov chain in an appropriately chosen phase space (a subset of a function space) that contains the zero function and is invariant for the (random) flow of the equation. Concerning this Markov chain, we prove the following main result (see Theorem 2.2): The Markov chain has a unique invariant measure. To prove this theorem, we present a construction assigning, to any invariant measure, a Gibbs measure for a 1D system with compact phase space and apply a version of Ruelle-Perron-Frobenius uniqueness theorem to the corresponding Gibbs system. We also discuss ergodic properties of the invariant measure and corresponding properties of the original randomly forced PDE.

Original language | English |
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Pages (from-to) | 291-330 |

Number of pages | 40 |

Journal | Communications in Mathematical Physics |

Volume | 213 |

Issue number | 2 |

Publication status | Published - Sep 2000 |