Heriot-Watt Mathematics Report Series
HWM00-3, 3 March 2000
Stochastic dissipative PDE's and Gibbs measures
S B Kuksin and A Shirikyan
Abstract
\begin{abstract}
We study a class of dissipative nonlinear PDE's forced by a random force
$\eta^\omega(t,x)$, 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
$$
\eta^\omega(t,x)=\sum_{k=-\infty}^{+\infty}\delta(t-kT)\eta_k(x),
$$
where the $\eta_k$'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~\ref{t2.2}):
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.
\end{abstract}
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Full text: http://www.ma.hw.ac.uk/~kuksin/
Publication
S B Kuksin, A N Smith
, Stochastic dissipative PDE's and Gibbs measures, Communications in Mathematical Physics, 213, 291-330, (2000).
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