Effects of Noise on Entropy Evolution

Abstract

We study the convergence properties of the conditional (Kullback-Leibler) entropy in stochastic systems. We have proved very general results showing that asymptotic stability is a necessary and sufficient condition for the monotone convergence of the conditional entropy to its maximal value of zero. Additionally we have made specific calculations of the rate of convergence of this entropy to zero in a one-dimensional situations, illustrated by Ornstein-Uhlenbeck and Rayleigh processes, higher dimensional situations, and a two dimensional Ornstein-Uhlenbeck process with a stochastically perturbed harmonic oscillator and colored noise as examples. We also apply our general results to the problem of conditional entropy convergence in the presence of dichotomous noise. In both the single dimensional and multidimensional cases we are to show that the convergence of the conditional entropy to zero is monotone and at least exponential. In the specific cases of the Ornstein-Uhlenbeck and Rayleigh processes as well as the stochastically perturbed harmonic oscillator and colored noise examples, we have the rather surprising result that the rate of convergence of the entropy to zero is independent of the noise amplitude.