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

This article establishes cutoff thermalization (also known as the cutoff phenomenon ) for a class of generalized Ornstein–Uhlenbeck systems ( X t ε ( x ) ) t ⩾ 0 with ε -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 ∞ / 0 -collapse of the renormalized Wasserstein distance from the current state to the equilibrium measure μ ε along a time window centered on a precise ε -dependent time scale 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 W p ( X t ε + r ε ( x ) , μ ε ) · ε - 1 → K · e - q r for any r ∈ R as ε → 0 for some spectral constants K , q > 0 and any 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 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 ε -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.