Stochastic Dissipative PDE's and Gibbs Measures
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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.
KeywordsPhase Space Markov Chain Invariant Measure Random Field Dirichlet Boundary
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