Basis Markov Partitions and Transition Matrices for Stochastic Systems

We analyze discrete-time dynamical systems subjected to an additive noise and their deterministic limit. In this work, we will introduce a notion by which a discrete-time stochastic system has something like a Markov partition for deterministic systems. For a chosen class of noise profiles, the Frobenius–Perron (FP) operator associated to the noisy system is exactly represented by a stochastic transition matrix of a finite size K. This feature allows us to introduce for these stochastic systems a basis Markov partition, defined herein, irrespectively of whether the deterministic system possesses a Markov partition or not. We show that in the deterministic limit, corresponding to K → ∞, the sequence of invariant measures of the noisy systems tends, in the weak sense, to the invariant measure of the deterministic system. Thus, by introducing a small additive noise one may approximate transition matrices and invariant measures of deterministic dynamical systems.