Wiener randomization on unbounded domains and an application to almost sure well-posedness of NLS

Bényi, Árpád; Oh, Tadahiro; Pocovnicu, Oana

doi:10.1007/978-3-319-20188-7_1

Cited by 72 publications

(230 citation statements)

References 58 publications

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“…We would like to stress again, however, that our reason for considering the randomization of the form (1.9) comes from its connection to time-frequency analysis. See also our previous papers [3] and [4].…”

Section: Contentsmentioning

confidence: 94%

“…When d = 4, the Hamiltonian is invariant under the scaling (1.2) and plays a crucial role in the global well-posedness theory. Indeed, Ryckman-Vişan [53] proved global well-posedness and scattering for the defocusing cubic NLS on R 4 . See also Vişan [60].…”

Section: Contentsmentioning

confidence: 99%

“…Almost simultaneously with our first paper [4], Lührmann-Mendelson [42] also considered a randomization of the form (1.9) (with cubes Q n being substituted by appropriately localized balls) in the study of NLW on R 3 . See Remark 1.6 below.…”

Section: Contentsmentioning

confidence: 99%

See 2 more Smart Citations

On the probabilistic Cauchy theory of the cubic nonlinear Schrödinger equation on ℝ^{𝕕}, 𝕕≥3

Bényi

Pocovnicu

2015

Trans. Amer. Math. Soc. Ser. B

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329

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Abstract. We consider the Cauchy problem of the cubic nonlinear Schrö-dinger equation (NLS) : i∂ t u + Δu = ±|u| 2 u on R d , d ≥ 3, with random initial data and prove almost sure well-posedness results below the scaling-critical regularity. More precisely, given a function on R d , we introduce a randomization adapted to the Wiener decomposition, and, intrinsically, to the so-called modulation spaces. Our goal in this paper is three-fold. (i) We prove almost sure local well-posedness of the cubic NLS below the scalingcritical regularity along with small data global existence and scattering. (ii) We implement a probabilistic perturbation argument and prove 'conditional' almost sure global well-posedness for d = 4 in the defocusing case, assuming an a priori energy bound on the critical Sobolev norm of the nonlinear part of a solution; when d = 4, we show that conditional almost sure global wellposedness in the defocusing case also holds under an additional assumption of global well-posedness of solutions to the defocusing cubic NLS with deterministic initial data in the critical Sobolev regularity. (iii) Lastly, we prove global well-posedness and scattering with a large probability for initial data randomized on dilated cubes.

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Section: Contentsmentioning

confidence: 94%

Section: Contentsmentioning

confidence: 99%

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On the probabilistic Cauchy theory of the cubic nonlinear Schrödinger equation on ℝ^{𝕕}, 𝕕≥3

Bényi

Pocovnicu

2015

Trans. Amer. Math. Soc. Ser. B

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329

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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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“…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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Wiener randomization on unbounded domains and an application to almost sure well-posedness of NLS

Cited by 72 publications

References 58 publications

On the probabilistic Cauchy theory of the cubic nonlinear Schrödinger equation on ℝ^{𝕕}, 𝕕≥3

On the probabilistic Cauchy theory of the cubic nonlinear Schrödinger equation on ℝ^{𝕕}, 𝕕≥3

Stability of invariant measures and continuity of the KdV flow

Randomization and the Gross–Pitaevskii Hierarchy

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