Invariant measure for the Schrödinger equation on the real line

Cacciafesta, Federico; Suzzoni, Anne-Sophie de

doi:10.1016/j.jfa.2015.04.021

Cited by 23 publications

(32 citation statements)

References 18 publications

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“…We refer the interested reader to the works [18,19,[28][29][30][31][32][33][34][35][36]61,[68][69][70]122,125,126,[130][131][132][133]144,[150][151][152][153][154][155][156][157][158]163], as well as to the expository works [27,166] and to the references therein. Furthermore, we note that the idea of randomization of the Fourier coefficients without the use of an invariant measure has also been applied in the context of the Navier-Stokes equations.…”

Section: Previously Known Resultsmentioning

confidence: 99%

Randomization and the Gross–Pitaevskii Hierarchy

Sohinger

Staffilani

2015

Arch Rational Mech Anal

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We study the Gross-Pitaevskii hierarchy on the spatial domain T 3 . By using an appropriate randomization of the Fourier coefficients in the collision operator, we prove an averaged form of the main estimate which is used in order to contract the Duhamel terms that occur in the study of the hierarchy. In the averaged estimate, we do not need to integrate in the time variable. An averaged spacetime estimate for this range of regularity exponents then follows as a direct corollary. The range of regularity exponents that we obtain is α > 3 4 . It was shown in our previous joint work with Gressman (J Funct Anal 266(7): 2014) that the range α > 1 is sharp in the corresponding deterministic spacetime estimate. This is in contrast to the non-periodic setting, which was studied by Klainerman and Machedon (Commun Math Phys 279(1): [169][170][171][172][173][174][175][176][177][178][179][180][181][182][183][184][185] 2008), where the spacetime estimate is known to hold whenever α ≥ 1. The goal of our paper is to extend the range of α in this class of estimates in a probabilistic sense. We use the new estimate and the ideas from its proof in order to study randomized forms of the Gross-Pitaevskii hierarchy. More precisely, we consider hierarchies similar to the Gross-Pitaevskii hierarchy, but in which the collision operator has been randomized. For these hierarchies, we show convergence to zero in low regularity Sobolev spaces of Duhamel expansions of fixed deterministic density matrices. We believe that the study of the randomized collision operators could be the first step in the understanding of a nonlinear form of randomization.

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Section: Previously Known Resultsmentioning

confidence: 99%

Randomization and the Gross–Pitaevskii Hierarchy

Sohinger

Staffilani

2015

Arch Rational Mech Anal

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“…The strategy for constructing invariant measures on compact manifolds, as inspired by [15] initially for a NLS, refined by Bourgain in [2], [3] and then followed by many authors (we mention at least [16], [19], and [20]), basically relies on applying a frequency truncation to reduce to a finite dimensional system, exploiting conservation of Lebesgue measure, which is a consequence of Liouville Theorem, and then proving uniform probabilistic estimates to remove truncates. The non compact case represents instead a much more challenging problem and not so many results are available in literature (see [1], [4], [12], [21], [8] and references therein); this represent in fact a very active branch of research.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Stability of invariant measures and continuity of the KdV flow

Cacciafesta

2016

Bull Braz Math Soc, New Series

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“…Lemma 3.2. Let Z L , Z L,1 , Z L,2 and Z L,3 defined respectively by (10), (11), (12) and (9). We have…”

Section: Estimatesmentioning

confidence: 99%

Invariance of Gibbs measures under the flows of Hamiltonian equations on the real line

Suzzoni

Cacciafesta

2019

Commun. Contemp. Math.

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We prove that the Gibbs measures ρ for a class of Hamiltonian equations written ason the real line are invariant under the flow of (1) in the sense that there exist random variables X(t) whose laws are ρ (thus independent from t) and such that t → X(t) is a solution to (1). Besides, for all t, X(t) is almost surely not in L 2 which provides as a direct consequence the existence of global weak solutions for initial data not in L 2 . The proof uses Prokhorov's theorem, Skorohod's theorem, as in the strategy in [8] and Feynman-Kac's integrals.

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Invariant measure for the Schrödinger equation on the real line

Cited by 23 publications

References 18 publications

Randomization and the Gross–Pitaevskii Hierarchy

Randomization and the Gross–Pitaevskii Hierarchy

Stability of invariant measures and continuity of the KdV flow

Invariance of Gibbs measures under the flows of Hamiltonian equations on the real line

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