Finite Time Dynamics and Finite Time Predictions for Stochastic and Deterministic Chaotic Systems

Probability theory deals with limit theorems, which consider limits (when time tends to infinity) of some functions (observables) on a sample space, or averages of these observables over an infinite time interval. But what is happening in a finite time or over finite time intervals? Such questions, important for virtually all applications, seem to be intractable mathematically (and generally sound unreasonable). For instance, equilibrium statistical mechanics deals with phase transitions (a number of equilibrium probability distributions/states) rather than time evolution, while nonequilibrium statistical mechanics is concerned with convergence of nonequilibrium states to equilibrium ones. Again, such processes occur on infinite time intervals. It turns out, however, that there are natural and reasonable questions about finite time dynamics of random and deterministic chaotic systems, which can be answered and, moreover, rigorously answered. This allows one to make predictions about a finite time evolution of such systems.