Igor Chueshov, Jinqiao Duan and Bjorn Schmalfuss
Determining functionals for random partial differential equations
(273K, ps file)
ABSTRACT. Determining functionals are tools to describe the finite
dimensional long-term dynamics of infinite dimensional dynamical
systems. There also exist several applications to infinite
dimensional {\em random} dynamical systems. In these applications the
convergence condition of the trajectories of an infinite
dimensional random dynamical system with respect to a finite set
of linear functionals is assumed to be either in mean or exponential
with
respect to the convergence almost surely. In contrast
to these ideas we introduce a convergence concept which is based
on the convergence in probability. By this ansatz we get rid of
the assumption of exponential convergence. In addition, setting
the random terms
to
zero we obtain usual deterministic results.
We apply our results to the 2D Navier - Stokes equations
forced by a white noise.