Mixing for dynamical systems driven by stationary noise

Geometric and Functional Analysis. 2025. Vol. 35. P. 1346–1399.

Kuksin S., Shirikyan A.

The paper deals with the problem of long-time asymptotic behaviour of solutions for classes of ODEs and PDEs, perturbed by stationary noises. The latter are not assumed to be δ-correlated in time, therefore the evolu- tion in question is not necessarily Markovian. We first prove an abstract result which implies the mixing for random dynamical systems satisfying appropriate dissipativity and controllability conditions. It is applicable to a large class of evolution equations, and we illustrate this on the ex- amples of a chain of anharmonic oscillators coupled to heat reservoirs, the 2d Navier–Stokes system, and a complex Ginzburg–Landau equation. Our results also apply to the general theory of random processes on the 1d lattice and allow one to get for them results related to Dobrushin’s theorems on reconstructing processes via their conditional distributions. The proof is based on an iterative construction with Newton’s quadratic approximation. It uses the method of Kantorovich functional, introduced earlier by the authors in the context of randomly forced PDEs, and some ideas used by them in the Markovian case to prove mixing with the help of controllability properties of an associated system.

Keywords: Mixing, Exponential mixing, stationary processes, Kantorovich functional, bounded random force