Tunnel Effect for Kramers-Fokker-Planck Type Operators: Return to Equilibrium and Applications

Hérau, Frédéric; Hitrik, Michael; Sjöstrand, Johannes

doi:10.1093/imrn/rnn057

Cited by 17 publications

(42 citation statements)

References 21 publications

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“…Now this result becomes a direct application of Theorem 11.6 to A := −P/h and we do not need any bounds on the resolvent in the region z > h ω. In [45,46] similar results were obtained for more general operators, for which we do not necessarily have any bound on the resolvent beyond a strip, and the proof was to use microlocal coercivity outside a compact set in slightly weighted L 2 -spaces. Again Theorem 11.6 would give some simplifications.…”

Section: Introductionsupporting

confidence: 58%

“…the study of the exponential decay of eigenfunctions. As an application we have a precise result on the return to equilibrium [46]. This has many similarities with older work on the tunnel effect for Schrödinger operators in the semi-classical limit by B. Helffer-Sjöstrand [42,43] and B. Simon [93] but for the Kramers-FokkerPlanck operator the problem is richer and more difficult since P is neither elliptic nor self-adjoint.…”

Section: Introductionmentioning

confidence: 62%

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Lecture notes : Spectral properties of non-self-adjoint operators

Sjöstrand¹

2011

Journées Équations Aux Dérivées Partielles

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RésuméCe texte contient une version légèrement completée de mon cours de 6 heures au colloque d'équations aux dérivées partielles à Évian-les-Bains en juin 2009. Dans la première partie on expose quelques résultats anciens et récents sur les opérateurs non-autoadjoints. La deuxième partie est consacrée aux résultats récents sur la distribution de Weyl des valeurs propres des opé-rateurs elliptiques avec des petites perturbations aléatoires. La partie III, en collaboration avec B. Helffer, donne des bornes explicites dans le théorème de Gearhardt-Prüss pour des semi-groupes. AbstractThis text contains a slightly expanded version of my 6 hour mini-course at the PDE-meeting in Évian-les-Bains in June 2009. The first part gives some old and recent results on non-self-adjoint differential operators. The second part is devoted to recent results about Weyl distribution of eigenvalues of elliptic operators with small random perturbations. Part III, in collaboration with B. Helffer, gives explicit estimates in the Gearhardt-Prüss theorem for semi-groups.

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Section: Introductionsupporting

confidence: 58%

Section: Introductionmentioning

confidence: 62%

Lecture notes : Spectral properties of non-self-adjoint operators

Sjöstrand¹

2011

Journées Équations Aux Dérivées Partielles

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“…Application: Null-controllability and observability of a chain of two oscillators coupled to two heat baths at each side This section is devoted to provide an application of the general results of null controllability and observability for accretive quadratic operators with zero singular spaces. This example given in [50,Section 4.3] comes from the series of works [18,19,20,27,28]. It is a model describing a chain of two oscillators coupled with two heat baths at each side.…”

Section: Proofs Of Null-controllability and Observability Of Hypoellimentioning

confidence: 99%

Null-controllability of hypoelliptic quadratic differential equations

Beauchard¹,

Pravda-Starov²

2018

Journal De L’École Polytechnique — Mathématiques

105

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We study the null-controllability of parabolic equations associated with a general class of hypoelliptic quadratic differential operators. Quadratic differential operators are operators defined in the Weyl quantization by complex-valued quadratic symbols. We consider in this work the class of accretive quadratic operators with zero singular spaces. These possibly degenerate non-selfadjoint differential operators are known to be hypoelliptic and to generate contraction semigroups which are smoothing in specific Gelfand-Shilov spaces for any positive time. Thanks to this regularizing effect, we prove by adapting the Lebeau-Robbiano method that parabolic equations associated with these operators are null-controllable in any positive time from control regions, for which nullcontrollability is classically known to hold in the case of the heat equation on the whole space. Some applications of this result are then given to the study of parabolic equations associated with hypoelliptic Ornstein-Uhlenbeck operators acting on weighted L 2 spaces with respect to invariant measures. By using the same strategy, we also establish the null-controllability in any positive time from the same control regions for parabolic equations associated with any hypoelliptic Ornstein-Uhlenbeck operator acting on the flat L 2 space extending in particular the known results for the heat equation or the Kolmogorov equation on the whole space.

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“…In the equilibrium case, a result analogous to Theorem 3.1 has been obtained in [12] when W is a Morse function with two local minima and one saddle point.…”

Section: Chain Of Oscillators Coupled To Heat Bathsmentioning

confidence: 82%

Tunnel effect and symmetries for non-selfadjoint operators

Hitrik¹

2014

Journées Équations Aux Dérivées Partielles

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Tunnel effect and symmetries for non-selfadjoint operators J. É. D. P. (2013), ExposéTunnel effect and symmetries for non-selfadjoint operators Michael Hitrik AbstractWe study low lying eigenvalues for non-selfadjoint semiclassical differential operators, where symmetries play an important role. In the case of the Kramers-Fokker-Planck operator, we show how the presence of certain supersymmetric and PT -symmetric structures leads to precise results concerning the reality and the size of the exponentially small eigenvalues in the semiclassical (here the low temperature) limit. This analysis also applies sometimes to chains of oscillators coupled to two heat baths, but when the temperatures of the baths are different, we show that the supersymmetric approach may break down. We also discuss PT -symmetric quadratic differential operators with real spectrum and characterize those that are similar to selfadjoint operators. This talk is based on joint works with

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Tunnel Effect for Kramers-Fokker-Planck Type Operators: Return to Equilibrium and Applications

Cited by 17 publications

References 21 publications

Lecture notes : Spectral properties of non-self-adjoint operators

Lecture notes : Spectral properties of non-self-adjoint operators

Null-controllability of hypoelliptic quadratic differential equations

Tunnel effect and symmetries for non-selfadjoint operators

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