Cutoff Thermalization for Ornstein–Uhlenbeck Systems with Small Lévy Noise in the Wasserstein Distance

Abstract

AbstractThis 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.