Cutoff thermalization for Ornstein-Uhlenbeck systems with small Lévy noise in the Wasserstein distance

Abstract

This article establishes cutoff thermalization (also known as the cutoff phenomenon) for a class of generalized Ornstein-Uhlenbeck systems $(X_t^{\epsilon }(x){)}_{t⩾0}$ with $\epsilon$-small additive Lévy noise and initial value $x$. The driving noise processes include Brownian motion, $\alpha$-stable Lévy 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 ${\mu }^{\epsilon }$ along a time window centered on a precise $\epsilon$- and $x$-dependent time scale $t_{\epsilon }^x$. In many interesting situations such as reversible (Lévy) 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 ${\mathcal{W}}_p(X_{t_{\epsilon }+r}^{\epsilon }(x),{\mu }^{\epsilon })\cdot {\epsilon }^{-1}\to K\cdot e^{-qr}$ as $\epsilon \to 0$ for any $r\in \mathbb{R}$, some spectral constants $K,q>0$ and any $p⩾1$ 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 $\epsilon$-small Brownian motion or $\alpha$-stable Lévy flights. Furthermore, we cover the highly degenerate case of a linear chain of oscillators in a generalized heat bath at low temperature.

47 pages

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