Time Scales and Exponential Trend to Equilibrium: Gaussian Model Problems

Abstract

We review results on the exponential convergence of multidimensional Ornstein-Uhlenbeck processes and discuss notions of characteristic time scales by means of concrete model systems. We focus, on the one hand, on exit time distributions and provide explicit expressions for the exponential rate of the distribution in the small-noise limit. On the other hand, we consider relaxation time scales of the process to its equilibrium measure in terms of relative entropy and discuss the connection with exit probabilities. Along these lines, we study examples which illustrate specific properties of the relaxation and discuss the possibility of deriving a simulation-based, empirical definition of slow and fast degrees of freedom which builds upon a partitioning of the relative entropy functional in connection with the observed relaxation behaviour.