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This article establishes cutoff thermalization (also known as the cutoff phenomenon) for a class of generalized Ornstein–Uhlenbeck systems $$(X^\varepsilon _t(x))_{t\geqslant 0}$$ ( X t ε ( x ) ) t ⩾ 0 with $$\varepsilon $$ ε -small additive Lévy noise and initial value x. The driving noise processes include Brownian motion, $$\alpha $$ 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 $$\infty /0$$ ∞ / 0 -collapse of the renormalized Wasserstein distance from the current state to the equilibrium measure $$\mu ^\varepsilon $$ μ ε along a time window centered on a precise $$\varepsilon $$ ε -dependent time scale $$\mathfrak {t}_\varepsilon $$ t ε . 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 $$\mathcal {W}_p(X^\varepsilon _{t_\varepsilon + r}(x), \mu ^\varepsilon ) \cdot \varepsilon ^{-1} \rightarrow K\cdot e^{-q r}$$ W p ( X t ε + r ε ( x ) , μ ε ) · ε - 1 → K · e - q r for any $$r\in \mathbb {R}$$ r ∈ R as $$\varepsilon \rightarrow 0$$ ε → 0 for some spectral constants $$K, q>0$$ K , q > 0 and any $$p\geqslant 1$$ 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 $$\mathcal {Q}$$ 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 $$\varepsilon $$ ε -small Brownian motion or $$\alpha $$ 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.

We consider a finite dimensional deterministic dynamical system with a global attractor A with a unique ergodic measure P concentrated on it, which is uniformly parametrized by the mean of the trajectories in a bounded set D containing A. We perturbe this dynamical system by a multiplicative heavy tailed Lévy noise of small intensity ε > 0 and solve the asymptotic first exit time and location problem from a bounded domain D around the attractor A in the limit of ε ց 0. In contrast to the case of Gaussian perturbations, the exit time has the asymptotically algebraic exit rate as a function of ε, just as in the case when A is a stable fixed point (see for instance [9,18,24]). In the small noise limit, we determine the joint law of the first time and the exit location from D c . As an example, we study the first exit problem from a neighbourhood of a stable limit cycle for the Van der Pol oscillator perturbed by multiplicative α-stable Lévy noise.

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Bibliografische Information der Deutschen NationalbibliothekDie Deutsche Nationalbibliothek verzeichnet diese Publikation in der Deutschen Nationalbibliografie; detaillierte bibliografische Daten sind im Internet über http://dnb.de abrufbar. Universitätsverlag AbstractWe introduce the notion of coupling distances on the space of Lévy measures in order to quantify rates of convergence towards a limiting Lévy jump diffusion in terms of its characteristic triplet, in particular in terms of the tail of the Lévy measure. The main result yields an estimate of the Wasserstein-Kantorovich-Rubinstein distance on path space between two Lévy diffusions in terms of the couping distances. We want to apply this to obtain precise rates of convergence for Markov chain approximations and a statistical goodness-of-fit test for lowdimensional conceptual climate models with paleoclimatic data.