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

TL;DR

This paper investigates the cut-off phenomenon and asymptotic mixing in multivariate linear processes, establishing conditions under which systems rapidly reach equilibrium across various stochastic models, including fractional and non-Gaussian processes.

Contribution

It provides new sufficient conditions for the cut-off phenomenon in a broad class of multivariate linear processes, extending previous results to non-Markovian and inhomogeneous cases.

Findings

Established conditions for cut-off in total variation and Wasserstein distances.

Applied results to fractional Ornstein--Uhlenbeck and related processes.

Demonstrated rapid mixing in complex stochastic systems.

Abstract

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 L\'evy 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.