Cut-off phenomenon and asymptotic mixing for multivariate general linear processes

The small noise cut-off phenomenon in continuous time and space has been studied in the recent literature for the linear and non-linear stable Langevin dynamics with additive Lévy drivers - understood as abrupt thermalization of the system along a particular time scale to its dynamical equilibrium - both for the total variation distance and the Wasserstein distance. The main result of this article establishes sufficient conditions for the window and profile cut-off phenomenon, which are flexible enough to cover the renormalized (non-Markovian) Ornstein–Uhlenbeck process driven by fractional Brownian motion and a large class of Gaussian and non-Gaussian, homogeneous and non-homogeneous drivers with (possible) finite second moments. The sufficient conditions are stated both for the total variation distance and the Wasserstein distance. Important examples are the multidimensional fractional Ornstein–Uhlenbeck process, the empirical sampling process of a fractional Ornstein–Uhlenbeck process, an Ornstein–Uhlenbeck processes driven by an Ornstein–Uhlenbeck process and the inhomogeneous Ornstein–Uhlenbeck process arising in simulated annealing.