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

Abstract This article establishes cutoff thermalization (also known as the phenomenon ) for a class of generalized Ornstein–Uhlenbeck systems $$(X^\varepsilon _t(x))_{t\geqslant 0}$$ (Xt?(x))t?0 with $$\varepsilon $$ xmlns:mml="http://www.w3.org/1998/Math/MathML">? -small additive Lévy noise and initial value x . The driving processes include Brownian motion, $$\alpha xmlns:mml="http://www.w3.org/1998/Math/MathML">? -stable flights, finite intensity compound Poisson processes, red noises, may be highly degenerate. Window is shown under mild generic assumptions; that is, we see an asymptotically sharp $$\infty /0$$ xmlns:mml="http://www.w3.org/1998/Math/MathML">?/0 -collapse renormalized Wasserstein distance from current state to equilibrium measure $$\mu ^\varepsilon xmlns:mml="http://www.w3.org/1998/Math/MathML">?? along time window centered on precise -dependent scale $$\mathfrak {t}_\varepsilon xmlns:mml="http://www.w3.org/1998/Math/MathML">t? In many interesting situations such reversible (Lévy) diffusions it possible prove existence explicit, universal, deterministic profile That data obtain stronger result $$\mathcal {W}_p(X^\varepsilon _{t_\varepsilon + r}(x), \mu \cdot \varepsilon ^{-1} \rightarrow K\cdot e^{-q r}$$ xmlns:mml="http://www.w3.org/1998/Math/MathML">Wp(Xt?+r?(x),??)·?-1?K·e-qr any $$r\in \mathbb {R}$$ xmlns:mml="http://www.w3.org/1998/Math/MathML">r?R 0$$ xmlns:mml="http://www.w3.org/1998/Math/MathML">??0 some spectral constants $$K, q>0$$ xmlns:mml="http://www.w3.org/1998/Math/MathML">K,q>0 $$p\geqslant 1$$ xmlns:mml="http://www.w3.org/1998/Math/MathML">p?1 whenever finite. this limit characterized by absence non-normal growth patterns in terms orthogonality condition computable family eigenvectors {Q}$$ xmlns:mml="http://www.w3.org/1998/Math/MathML">Q Precise error bounds are given. Using these results, provides complete discussion classical linear oscillator friction subject motion or flights. Furthermore, cover degenerate case chain oscillators heat bath at low temperature.