Modulation spaces, Wiener amalgam spaces, and Brownian motions

Bényi, Árpád; Oh, Tadahiro

doi:10.1016/j.aim.2011.07.023

Cited by 55 publications

(135 citation statements)

References 48 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: 99%

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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119

322

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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: 99%

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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119

322

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“…After posting this paper on ArXiv,Árpád Bényi and Tadahiro Oh kindly pointed out that there is some overlap with their paper [1], in which similar techniques are used to characterize the exponents (s, p, q) for which a Brownian motion on T is in B s p,q (T) and inb s p,q (T). In fact, in dimension d = 1 some (but not all) of our results could alternatively be deduced from theirs.…”

Section: Introductionmentioning

confidence: 99%

Regularity of Gaussian white noise on the d-dimensional torus

Veraar

2011

Banach Center Publ.

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“…The stochastic convolution, when measured in terms of its interaction representation S(−t)Ψ(t), then gains one temporal regularity and thus has temporal regularity b < 1 − 1 q . This is precisely the regularity of the Brownian motion measured in the Fourier-Lebesgue spaces; see [3]. The next lemma tells us the regularity of the stochastic convolution with respect to the X s,b p,q -spaces.…”

Section: On the Stochastic Convolutionmentioning

confidence: 91%

Stochastic Nonlinear Schrödinger Equation With Almost Space–time White Noise

Forlano¹,

Oh²,

Wang³

2019

J. Aust. Math. Soc.

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We study the stochastic cubic nonlinear Schrödinger equation (SNLS) with an additive noise on the one-dimensional torus. In particular, we prove local well-posedness of the (renormalized) SNLS when the noise is almost space–time white noise. We also discuss a notion of criticality in this stochastic context, comparing the situation with the stochastic cubic heat equation (also known as the stochastic quantization equation).

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Modulation spaces, Wiener amalgam spaces, and Brownian motions

Cited by 55 publications

References 48 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

Regularity of Gaussian white noise on the d-dimensional torus

Stochastic Nonlinear Schrödinger Equation With Almost Space–time White Noise

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