Cut-off and exit from metastability: two sides of the same coin

Bertoncini, Olivier; Barrera, Javiera; Fernández, Roberto

doi:10.1016/j.crma.2008.04.007

“…At this point we refrain from giving a full account on the mathematical literature on the cutoff phenomenon and refer to the overview article [41] and the introduction of [16]. Standard references in the mathematics literature on the cutoff phenomenon for discrete time and space include [1,3,4,9,15,18,32,33,40,42,[74][75][76][77][78]80,103,109]. As introductory texts on the cutoff phenomenon in discrete time and space we recommend [68] and Chapter 18 in the monograph [78].…”

Section: Introductionmentioning

confidence: 99%

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

Barrera

¹

,

Högele

²

,

Pardo

³

2021

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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.

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“…Proposition 1. Let m : Ξ → R be the function defined in (1). Under Assumption 1, for every Borel measurable A ⊆ Ξ it holds that…”

Section: Assumption 1 the Functionmentioning

confidence: 99%

“…Indeed, the conditions are quite natural. Condition (1) states that the time spent in any state is asymptotically negligible compared to the K dominating states, while on the other hand (2) asks that the asymptotic time spent in the dominating states be comparable. Condition (1) cannot be relaxed in general as will be explained in Remark 4.…”

Section: Emerging Dominancementioning

confidence: 99%

Self-Switching Markov Chains: emerging dominance phenomena

Gallo

¹

,

Iacobelli

²

,

Ost

³

et al. 2020

Preprint

0

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In many dynamical systems in nature, the law of the dynamics changes along with the temporal evolution of the system. These changes are often associated with the occurrence of certain events. The timing of occurrence of these events depends, in turn, on the trajectory of the dynamical system itself, making the dynamics of the system and the timing of a change in the dynamics strongly coupled. Naturally, trajectories that take longer to satisfy the event will last longer. Therefore, we expect to observe more frequently the dominant dynamics, the ones that take longer to change in the long run. This article proposes a Markov chain model, called Self-Switching Markov Chain (SSMC), in which the emergence of dominant dynamics can be rigorously addressed. We present conditions and scaling in the SSMC under which we observe with probability one only the subset of dominant dynamics, and we characterize these dominant dynamics. Furthermore, we show that the switching between dynamics exhibits metastability-like property.

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“…For instance we refer to Diaconis [30], Martínez and Ycart [54] and Levin et al [53] for the Markov chain setting, Chen and Saloff-Coste [25] considered some ergodic Markov processes, Lachaud [50] and Barrera [5] for the case of the Ornstein-Uhlenbeck processes driven by a Brownian motion, to name but a few. Further standard texts on the cutoff phenomenon include [1,3,13,15,11,14,17,18,25,29,31,32,51,52,53,55,74,79] and the references therein. The newest developments in this active field of research are found in the recent publications [16,19,20,21,24,49,40,41].…”

Section: Introductionmentioning

confidence: 99%

The cutoff phenomenon in total variation for nonlinear Langevin systems with small layered stable noise

Barrera

¹

,

Högele

²

,

Pardo

³

2021

Electron. J. Probab.

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This paper provides an extended case study of the cutoff phenomenon for a prototypical class of nonlinear Langevin systems with a single stable state perturbed by an additive pure jump Lévy noise of small amplitude ε > 0, where the driving noise process is of layered stable type. Under a drift coercivity condition the associated family of processes X ε turns out to be exponentially ergodic with equilibrium distribution µ ε in total variation distance which extends a result from [60] to arbitrary polynomial moments.The main results establish the cutoff phenomenon with respect to the total variation, under a sufficient smoothing condition of Blumenthal-Getoor index α > 3 2 . That is to say, in this setting we identify a deterministic time scale t cut ε satisfying t cut ε → ∞, as ε → 0, and a respective time window, t cut ε ± o(t cut ε ), during which the total variation distance between the current state and its equilibrium µ ε essentially collapses as ε tends to zero. In addition, we extend the dynamical characterization under which the latter phenomenon can be described by the convergence of such distance to a unique profile function first established in [9] to the Lévy case for nonlinear drift. This leads to sufficient conditions, which can be verified in examples, such as gradient systems subject to small symmetric α-stable noise for α > 3 2 . The proof techniques differ completely from the Gaussian case due to the absence of a respective Girsanov transform which couples the nonlinear equation and the linear approximation asymptotically even for short times.

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Cut-off and exit from metastability: two sides of the same coin

Cited by 13 publications

References 17 publications

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

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

Self-Switching Markov Chains: emerging dominance phenomena

The cutoff phenomenon in total variation for nonlinear Langevin systems with small layered stable noise

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