Noise and conditional entropy evolution

We study the convergence properties of the conditional (Kullback–Leibler) entropy in stochastic systems. We have proved 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 situation, 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 one-dimensional and multidimensional cases we 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 obtain exact formulae for the temporal evolution of the conditional entropy starting from a concrete initial distribution. The rather surprising result in this case is that the rate of convergence of the entropy to zero is independent of the noise amplitude.