Stochastic Dissipative PDE's and Gibbs Measures |
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Authors: | Sergei Kuksin Armen Shirikyan |
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Affiliation: | Department of Mathematics, Heriot-Watt University, Edinburgh EH14 4AS, Scotland, UK.?E-mail: S.B.Kuksin@ma.hw.ac.uk; A.Shirikyan@ma.hw.ac.uk, GB
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Abstract: | We study a class of dissipative nonlinear PDE's forced by a random force ηomega( 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 t and stationary, short-correlated in time t. In this paper, we confine ourselves to “kick forces” of the form where the η 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. Received: 24 January 2000 / Accepted: 17 February 2000 |
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