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 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 \(\mu ^\varepsilon \) along a time window centered on a precise \(\varepsilon \)-dependent time scale \(\mathfrak {t}_\varepsilon \). 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 \(\mathcal {W}_p(X^\varepsilon _{t_\varepsilon + r}(x), \mu ^\varepsilon ) \cdot \varepsilon ^{-1} \rightarrow K\cdot e^{-q r}\) for any \(r\in \mathbb {R}\) as \(\varepsilon \rightarrow 0\) for 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 Lévy flights. Furthermore, we cover the highly degenerate case of a linear chain of oscillators in a generalized heat bath at low temperature.

本文为一类广义 Ornstein-Uhlenbeck 系统\((X^\varepsilon _t(x))_{t\geqslant 0}\)与\(\varepsilon \)建立了截止热化（也称为截止现象） -小的附加 Lévy 噪声和初始值x。驱动噪声过程包括布朗运动、\(\alpha \)- stable Lévy 飞行、有限强度复合泊松过程和红噪声，并且可能是高度退化的。窗口截止热化显示在温和的通用假设下；也就是说，我们看到一个渐近尖锐的\(\infty /0\)- 从当前状态到平衡测量值\(\mu ^\varepsilon \)的重整化 Wasserstein 距离沿着以精确\(\varepsilon \)依赖的时间尺度为中心的时间窗口的崩溃\(\mathfrak {t}_ \varepsilon \)。在许多有趣的情况下，例如可逆 (Lévy) 扩散，可以证明明确的、通用的、确定性的截止热化曲线的存在。也就是说，对于通用初始数据x我们得到更强的结果\(\mathcal {W}_p(X^\varepsilon _{t_\varepsilon + r}(x), \mu ^\varepsilon ) \cdot \varepsilon ^{ -1} \rightarrow K\cdot e^{-qr}\)对于任何\(r\in \mathbb {R}\)作为\(\varepsilon \rightarrow 0\)对于一些谱常数\(K, q>0\)和任何\(p\geqslant 1\)只要距离是有限的。这个限制的存在的特点是在\(\mathcal {Q}\)的广义特征向量的可计算族的正交条件方面不存在非正态增长模式。给出了精确的误差界限。使用这些结果，本文对经典线性振荡器的截止现象进行了完整的讨论，摩擦受\(\varepsilon \) -小布朗运动或\(\alpha \)- 稳定的 Lévy 航班。此外，我们涵盖了低温广义热浴中线性振荡器链的高度简并情况。