Determining functionals for random partial differential equations

Nonlinear Diff. Eqns. Appl. (NoDEA) 10(2003), 431-454. Determining functionals are tools to describe the finite dimensional longterm dynamics of infinite dimensional dynamical systems. There also exist several applications to infinite dimensional 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. on leave from Department of Mechanics and Mathematics, Kharkov University, 310077 Kharkov, Ukraine