Cutoff Thermalization for Ornstein-Uhlenbeck Systems with Small Levy Noise in the Wasserstein Distance

This article establishes cutoff thermalization (also knownas the cutoff phenomenon) foraclass of generalized Ornstein-Uhlenbeck systems (X-t(epsilon) (x))(t >= 0) with e-small additive Levy noise and initial value x. The driving noise processes include Brownian motion, alpha-stable Levy 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 infinity/0-collapse of the renormalized Wasserstein distance from the current state to the equilibrium measure mu epsilon along a time window centered on a precise epsilon-dependent time scale t(epsilon). In many interesting situations such as reversible (Levy) 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 epsilon+r(epsilon) (x), mu(epsilon)).e(-1)-> K.e(-qr) for any r is an element of R as epsilon -> 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 nonnormal 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 epsilon-small Brownian motion or alpha-stable Levy flights. Furthermore, we cover the highly degenerate case of a linear chain of oscillators in a generalized heat bath at low temperature.