A Note on Harris’ Ergodic Theorem, Controllability and Perturbations of Harmonic Networks

Raquépas, Renaud

doi:10.1007/s00023-018-0740-0

Cited by 10 publications

(16 citation statements)

References 18 publications

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“…Here L 1 , L n are one dimensional independent Lévy processes satisfying Hypothesis 2.1 for some p > 0. By Section 4.1 in [92] Q satisfies Hypothesis 2.2. Consequently by Theorem 3.4 the system exhibits window cutoff thermalization for any initial condition x = 0.…”

Section: Window Cutoff Thermalization For the Linear Chain Of Oscillatorsmentioning

confidence: 94%

“…Our results cover the setting of Jacobi chains of n oscillators with nearest neighbor interactions coupled to heat baths at its two ends, as discussed in Section 4.1 in [92] and Section 4.2 in [69]. For the sake of simplicity we show window cutoff thermalization for n oscillators with the Hamiltonian…”

Section: Window Cutoff Thermalization For the Linear Chain Of Oscillatorsmentioning

confidence: 97%

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

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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Section: Window Cutoff Thermalization For the Linear Chain Of Oscillatorsmentioning

confidence: 94%

Section: Window Cutoff Thermalization For the Linear Chain Of Oscillatorsmentioning

confidence: 97%

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

2021

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“…Therefore, the proof of Proposition 1.1 still holds, with minor differences, when we consider the following b.c. as well (free in a sense): (iv) A convergence to equilibrium in total variation norm for a similar small perturbation of the harmonic oscillator chain, has been shown recently in [32]. There, a version of Harris' ergodic Theorem was applied making it possible to treat more general cases of the oscillator chain with different kind of noises, as well.…”

Section: Remark 51 (I)mentioning

confidence: 90%

Quantitative Rates of Convergence to Non-equilibrium Steady State for a Weakly Anharmonic Chain of Oscillators

Menegaki

2020

J Stat Phys

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We study a 1-dimensional chain of N weakly anharmonic classical oscillators coupled at its ends to heat baths at different temperatures. Each oscillator is subject to pinning potential and it also interacts with its nearest neighbors. In our set up both potentials are homogeneous and bounded (with N dependent bounds) perturbations of the harmonic ones. We show how a generalised version of Bakry-Emery theory can be adapted to this case of a hypoelliptic generator which is inspired by Baudoin (J Funct Anal 273(7):2275-2291, 2017). By that we prove exponential convergence to non-equilibrium steady state in Wasserstein-Kantorovich distance and in relative entropy with quantitative rates. We estimate the constants in the rate by solving a Lyapunov-type matrix equation and we obtain that the exponential rate, for the homogeneous chain, has order bigger than N −3. For the purely harmonic chain the order of the rate is in [N −3 , N −1 ]. This shows that, in this set up, the spectral gap decays at most polynomially with N. Keywords Chain of oscillators • Spectral gap • Hypocoercivity • Hypoellipticity • Functional inequalities • Nonequilibrium steady states • Heat bath • Exponential convergence N i=0 U int (q i+1 − q i), Communicated by Stefano Olla.

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“…In this section, we apply the Main Theorem to the Galerkin approximations of pdes and to stochastically driven quasi-harmonic networks. For the Galerkin approximations we give a detailed derivation of the controllability conditions and in the case of the networks we appeal to the results obtained in [Raq19]. Before we do so, we briefly discuss the solid controllability assumption (C3).…”

Section: Applicationsmentioning

confidence: 99%

“…In the paper [Shi17], a general approach based on controllability and a coupling argument is given for a study of dynamical systems on compact metric spaces subject to a degenerate noise: under the controllability assumptions (C2) and (C3) and a decomposability assumption on the noise, exponential mixing in the total-variation metric is established. This approach can be carried to problems on a non-compact space, provided a dissipativity of the type of (C1) holds; see [Raq19] for a study of networks of quasi-harmonic oscillators. The class of decomposable noises includes, but is not limited to, Gaussian measures.…”

mentioning

confidence: 99%

Exponential mixing under controllability conditions for SDEs driven by a degenerate Poisson noise

Nersesyan,

Raquépas

2019

Preprint

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We prove existence and uniqueness of the invariant measure and exponential mixing in the total-variation norm for a class of stochastic differential equations driven by degenerate compound Poisson processes. In addition to mild assumptions on the distribution of the jumps for the driving process, the hypotheses for our main result are that the corresponding control system is dissipative, approximately controllable and solidly controllable. The solid controllability assumption is weaker than the well-known parabolic Hörmander condition and is only required from a single point to which the system is approximately controllable. Our analysis applies to Galerkin projections of stochastically forced parabolic partial differential equations with asymptotically polynomial nonlinearities and to networks of quasi-harmonic oscillators connected to different Poissonian baths.

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A Note on Harris’ Ergodic Theorem, Controllability and Perturbations of Harmonic Networks

Cited by 10 publications

References 18 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

Quantitative Rates of Convergence to Non-equilibrium Steady State for a Weakly Anharmonic Chain of Oscillators

Exponential mixing under controllability conditions for SDEs driven by a degenerate Poisson noise

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