Year: 2026
Type: preprint
Abstract: We study the mixing properties of discrete-time and continuous-time dissipative dynamical 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 random 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 metric 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.
Source: arXiv (Cornell University)
Authors: Gao, Peng, Kuksin, Sergei
Institutions: National Research University Higher School of Economics, Peoples' Friendship University of Russia, Northeast Normal University, Centre National de la Recherche Scientifique, Université Paris Cité, Steklov Mathematical Institute, Institut de Mathématiques de Jussieu-Paris Rive Gauche, Sorbonne Université
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7167729049
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