论文标题
在Wasserstein距离内具有小lévy噪声的Ornstein-Uhlenbeck系统的截止热化
Cutoff thermalization for Ornstein-Uhlenbeck systems with small Lévy noise in the Wasserstein distance
论文作者
论文摘要
本文为一类通用的Ornstein-uhlenbeck Systems $（X^\ VAREPSILON_T（X））_ {t \ geqslant 0} $，带有$ \ varepsilon $ -small添加剂lévylévylévy噪声和初始值$ x $。驾驶噪声过程包括布朗运动，$α$稳定的莱维飞行，有限强度复合泊松过程和红色噪声，并且可能是高度退化的。窗口截止的热化显示在温和的通用假设下；也就是说，我们看到重新归一化的Wasserstein的渐近清晰$ \ Infty/0 $ collapse从当前状态到平衡度量$μ^\ varepsilon $沿着以精确的$ \ \ \ \ \ varepsilon $ - 和$ x $ x $ depperional-depperional-dippertiment-depperional-deppertent-depperted $ t_ $ t_ \ varepsilon^$。在许多有趣的情况下，例如可逆（Lévy）扩散，可以证明存在明确，通用，确定性的截止热化曲线。也就是说，对于通用初始数据$ x $，我们获得更强的结果$ \ Mathcal {w} _p（x^\ varepsilon_ {t_ \ varepsilon + r}（x）（x），μ^\ varepsilon），μ^\ varepsilon） $ \ varepsilon \ rightarrow 0 $对于任何$ r \ in \ mathbb {r} $，某些频谱常数$ k，q> 0 $和任何$ p \ geqslant 1 $，只要距离是有限的。该限制的存在的特征是在$ \ Mathcal {q} $的可计算的广义特征向量家族上，就正交性条件没有非正态生长模式。给出了精确的误差范围。使用这些结果，本文对经典的线性振荡器的截止现象进行了完整的讨论，其摩擦可能会受到$ \ varepsilon $ -small Brownian Motion或$α$稳定的Lévy飞行。此外，我们涵盖了在低温下广义热浴中振荡器线性链的高度退化情况。
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, $α$-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 $μ^\varepsilon$ along a time window centered on a precise $\varepsilon$- and $x$-dependent time scale $t_\varepsilon^x$. In many interesting situations such as reversible (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 $\mathcal{W}_p(X^\varepsilon_{t_\varepsilon + r}(x), μ^\varepsilon) \cdot \varepsilon^{-1} \rightarrow K\cdot e^{-q r}$ 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 $α$-stable Lévy flights. Furthermore, we cover the highly degenerate case of a linear chain of oscillators in a generalized heat bath at low temperature.
