Time Averages of Markov Processes and Applications to Two-Timescale Problems

Abstract

We show a decomposition into the sum of a martingale and a deterministic quantity for time averages of the solutions to non-autonomous SDEs and for discrete-time Markov processes. In the SDE case the martingale has an explicit representation in terms of the gradient of the associated semigroup or transition operator. We show how the results can be used to obtain quenched Gaussian concentration inequalities for time averages and to provide insights into the Averaging principle for two-timescale processes.