Prof. Mark Freidlin
Long time behavior of a dynamical system, evolutionary PDE, or a stochastic process, under certain assumptions, can be described by the invariant measure corresponding to the initial condition. If we consider perturbations of such a system, deterministic or stochastic, the long time behavior of the perturbed system can be described as a motion on the simplex of invariant measures of the original system. This motion has a limit as the perturbation tends to zero. The limiting motion on the simplex can be described by a generalized averaging principle or by large deviations principle. In particular, long time behavior of perturbed system can be stochastic even in pure deterministic case (due to instabilities in the original system), and can have strong deterministic components in the case of random perturbations (due to large deviations).