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

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

TL;DR

This work develops a comprehensive framework for the cut-off phenomenon in mixed linear stochastic systems with small noise, encompassing non-Markovian drivers such as fractional Brownian motion. By proving window and profile cut-off results in both total variation and renormalized Wasserstein distances for general linear processes X^{\varepsilon}_t = e^{-\Lambda t} x + \varepsilon S_t, the authors unify a broad class of models, including univariate and multivariate fractional OU processes, stochastic convolutions, and generalized OU dynamics. A key contribution is the Hartman--Grobman-based asymptotic description of the deterministic decay and the omega-limit set, together with explicit cut-off time scales t_{\varepsilon} and universal or semi-universal cut-off profiles, which are then illustrated via diverse examples such as averaging from fractional OU sampling, non-homogeneous and iterated OU schemes, and integrated OU processes. The results highlight how TV and Wasserstein distances capture different aspects of convergence and provide practical criteria for verifying cut-off in complex non-Markovian and non-Gaussian settings, with potential implications for simulated annealing, sampling, and stochastic modeling. The paper thus broadens the applicability of cut-off analysis to a wide spectrum of linear systems with irregular noise structures while preserving a clear, quantitative description of the approach to equilibrium.

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.

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

TL;DR

This work develops a comprehensive framework for the cut-off phenomenon in mixed linear stochastic systems with small noise, encompassing non-Markovian drivers such as fractional Brownian motion. By proving window and profile cut-off results in both total variation and renormalized Wasserstein distances for general linear processes X^{\varepsilon}_t = e^{-\Lambda t} x + \varepsilon S_t, the authors unify a broad class of models, including univariate and multivariate fractional OU processes, stochastic convolutions, and generalized OU dynamics. A key contribution is the Hartman--Grobman-based asymptotic description of the deterministic decay and the omega-limit set, together with explicit cut-off time scales t_{\varepsilon} and universal or semi-universal cut-off profiles, which are then illustrated via diverse examples such as averaging from fractional OU sampling, non-homogeneous and iterated OU schemes, and integrated OU processes. The results highlight how TV and Wasserstein distances capture different aspects of convergence and provide practical criteria for verifying cut-off in complex non-Markovian and non-Gaussian settings, with potential implications for simulated annealing, sampling, and stochastic modeling. The paper thus broadens the applicability of cut-off analysis to a wide spectrum of linear systems with irregular noise structures while preserving a clear, quantitative description of the approach to equilibrium.

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.

Paper Structure

This paper contains 16 sections, 12 theorems, 92 equations.

Table of Contents

  1. Introduction
  2. Univariate fractional Ornstein--Uhlenbeck process
  3. Cut-off convergence for general linear processes in law
  4. Multivariate cut-off phenomenon in the total variation distance
  5. Multivariate cut-off phenomenon in the Wasserstein distance
  6. Examples
  7. Linear equations and convolution processes $S$
  8. Recovering some existing results
  9. Average process from sampling a fractional Ornstein--Uhlenbeck processes
  10. Generalized Ornstein--Uhlenbeck processes
  11. Iterated Gaussian Ornstein--Uhlenbeck processes
  12. Non-homogeneous example
  13. Ex-centric integrated Ornstein--Uhlenbeck process
  14. Proofs of the main results
  15. Basic general properties of the total variation distance
  16. ...and 1 more sections

Key Result

Lemma 2.1

Let $H\in (0,1)$ be fixed. For all $t>0$, the random variable $X^{\varepsilon,H}_t(x)$ has Normal distribution with mean and variance where and $\Gamma$ denotes the usual Gamma function. Moreover, as $t\to \infty$ the law of the random variable $X^{\varepsilon,H}_t(x)$ converges (in the total variation distance and in the Wasserstein distance of any order $p\geq 1$) to a random variable $X^{\va

Theorems & Definitions (27)

  • Lemma 2.1: Ergodicity
  • Theorem 2.2: Cut-off convergence in the total variation distance
  • Remark 2.3: Total variation profile in terms of the error function
  • proof : Proof of Theorem \ref{['th:tv']} (Cut-off convergence in the total variation distance)
  • Theorem 2.4: Cut-off convergence in the Wasserstein distance of order $p\geq 1$
  • Remark 2.5: Wasserstein profile for $p\geq 1$ in terms of the exponential function
  • proof : Proof of Theorem \ref{['th:Wp']} (Cut-off convergence in the Wasserstein distance of order $p\geq 1$)
  • Lemma 3.1: Hartman–Grobman asymptotics
  • Theorem 3.2: Window cut-off convergence in the total variation distance
  • Remark 3.3: Growth of the function $\sigma=(\sigma_t)_{t\geq 0}$ at infinity
  • ...and 17 more