Dynamical entropy for systems with stochastic perturbation

Dynamics of deterministic systems perturbed by random additive noise is characterized quantitatively. Since for such systems the Kolmogorov-Sinai (KS) entropy diverges if the diameter of the partition tends to zero, we analyze the difference between the total entropy of a noisy system and the entropy of the noise itself. We show that this quantity is finite and non-negative and we call it the dynamical entropy of the noisy system. In the weak noise limit this quantity is conjectured to tend to the KS entropy of the deterministic system. In particular, we consider one-dimensional systems with noise described by a finite-dimensional kernel for which the Frobenius-Perron operator can be represented by a finite matrix.