Cutoff thermalization for Ornstein-Uhlenbeck systems with small L\'evy noise in the Wasserstein distance

TL;DR

This paper proves the cutoff thermalization phenomenon for a broad class of Ornstein-Uhlenbeck systems driven by small Le9vy noise, providing explicit profiles and error bounds, with applications to oscillators and heat baths.

Contribution

It establishes the cutoff phenomenon for generalized Ornstein-Uhlenbeck systems with small Le9vy noise, including explicit profiles and conditions for universality.

Findings

Sharp asymptotic Wasserstein distance collapse at cutoff time

Existence of explicit universal thermalization profiles

Application to linear oscillators and heat bath models

Abstract

This article establishes cutoff thermalization (also known as the cutoff phenomenon) for a class of generalized Ornstein-Uhlenbeck systems $(X^\varepsilon_t(x))_{t\geqslant 0}$ with $\varepsilon$-small additive L\'evy noise and initial value $x$. The driving noise processes include Brownian motion, $\alpha$-stable L\'evy flights, finite intensity compound Poisson processes, and red noises, and may be highly degenerate. Window cutoff thermalization is shown under mild generic assumptions; that is, we see an asymptotically sharp $\infty/0$-collapse of the renormalized Wasserstein distance from the current state to the equilibrium measure $\mu^\varepsilon$ along a time window centered on a precise $\varepsilon$- and $x$-dependent time scale $t_\varepsilon^x$. In many interesting situations such as reversible (L\'evy) diffusions it is possible to prove the existence of an explicit, universal, deterministic cutoff thermalization profile. That is, for generic initial data $x$ we obtain the stronger result $\mathcal{W}_p(X^\varepsilon_{t_\varepsilon + r}(x), \mu^\varepsilon) \cdot \varepsilon^{-1} \rightarrow K\cdot e^{-q r}$ as $\varepsilon \rightarrow 0$ for any $r\in \mathbb{R}$, some spectral constants $K, q>0$ and any $p\geqslant 1$ whenever the distance is finite. The existence of this limit is characterized by the absence of non-normal growth patterns in terms of an orthogonality condition on a computable family of generalized eigenvectors of $\mathcal{Q}$. Precise error bounds are given. Using these results, this article provides a complete discussion of the cutoff phenomenon for the classical linear oscillator with friction subject to $\varepsilon$-small Brownian motion or $\alpha$-stable L\'evy flights. Furthermore, we cover the highly degenerate case of a linear chain of oscillators in a generalized heat bath at low temperature.