Mixing in stochastic dynamical systems with stationary noise

The asymptotic behaviour of trajectories of stochastic dynamical systems with white noise has been studied in considerable depth both in the finite- and infinite-dimensional cases. As is well known, in this situation the system has a unique globally stable state, provided that the transition function of the Markov process generated by the system has some properties of regularity and recurrence: see [1], [5], and [2]. The aim of this note is to announce some recent results in the case when, in place of white noise, a stochastic dynamical system is driven by Markovian or stationary noise. The reader can find the proofs of theorems below in [3] and [4]. For the simplicity of presentation we limit ourselves to the case of finite-dimensional phase space. © 2024 Russian Academy of Sciences, Steklov Mathematical Institute of RAS.