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Cutoff thermalization for Ornstein-Uhlenbeck systems with small Lévy noise in the Wasserstein distance. (English) Zbl 1473.60067

Summary: 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 \(\mathcal{W}_p(X^\varepsilon_{t_\varepsilon+r}(x),\mu^\varepsilon) \cdot\varepsilon^{-1}\rightarrow K\cdot e^{-qr}\) 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.

MSC:

60G15 Gaussian processes
60G55 Point processes (e.g., Poisson, Cox, Hawkes processes)
60J65 Brownian motion

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