Long-time behaviour of dynamical systems driven by bounded mixing noises

Dynamical Systems. 2026.

Kuksin S.

We study the mixing properties of discrete-time and continuous-time dissipative dynam-
ical systems driven by bounded mixing random forces. The continuous-time systems are
reduced to discrete-time random dynamical systems generated by time-one maps, so that
the main analysis is carried out in the discrete setting. We introduce a class of mixing ran-
dom forcings whose regular conditional distributions with respect to the past satisfy natural
regularity, recurrence, and non-degeneracy assumptions, extending the framework previously
developed for more restrictive classes of processes in a paper by Kuksin-Shirikyan in GAFA
(2025). Under a linearised controllability assumptions on the system, we prove exponential
mixing in the total variation metric for finite-dimensional phase spaces. We then establish
an infinite-dimensional counterpart yielding exponential mixing in the dual-Lipschitz met-
ric under suitable amendments of restrictions on the system and the random forcing. Our
approach is based on lifting the dynamics to an appropriate Markov process on an infinite-
dimensional history space and applying a Doeblin coupling argument through the method of
Kantorovich functional. As applications, we derive exponential mixing for a broad class of
ordinary differential equations driven by mixing random processes with bounded continuous
trajectories. As an application of our result to PDEs we discuss the randomly perturbed
primitive equations of atmospheric dynamics.

Keywords: dynamical systems