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arXiv · 2502.07911

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

Gerardo Barrera·Michael A. Högele·Pauliina Ilmonen·Lauri Viitasaari
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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 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.

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Gerardo Barrera, Michael A. Högele, Pauliina Ilmonen, Lauri Viitasaari. 2025-02-11. Cut-off phenomenon and asymptotic mixing for multivariate general linear processes. https://arxiv.org/abs/2502.07911

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