Stochastic Nonlinear Schrödinger Equation With Almost Space–time White Noise

Forlano, Justin; Oh, Tadahiro; Wang, Yuzhao

doi:10.1017/s1446788719000156

Cited by 27 publications

(25 citation statements)

References 46 publications

(95 reference statements)

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“…For dispersive equations including the quadratic SNLW, the scaling analysis as above does not seem to provide any useful insight, 19 unless appropriate integrability conditions are incorporated. See, for example, [25] for a discussion in the case of the stochastic nonlinear Schrödinger equation. 20 Remark 1.13.…”

Section: Stochastic Nonlinear Heat Equationmentioning

confidence: 99%

Comparing the stochastic nonlinear wave and heat equations: a case study

Oh¹,

Okamoto

2021

Electron. J. Probab.

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We study the two-dimensional stochastic nonlinear wave equation (SNLW) and stochastic nonlinear heat equation (SNLH) with a quadratic nonlinearity, forced by a fractional derivative (of order α > 0) of a space-time white noise. In particular, we show that the well-posedness theory breaks at α = 1 2 for SNLW and at α = 1 for SNLH. This provides a first example showing that SNLW behaves less favorably than SNLH. (i) As for SNLW, Deya (2020) essentially proved its local well-posedness for 0 < α < 1 2 . We first revisit this argument and establish multilinear smoothing of order 1 4 on the second order stochastic term in the spirit of a recent work by Gubinelli, Koch, and Oh (2018). This allows us to simplify the local well-posedness argument for some range of α. On the other hand, when α ≥ 1 2 , we show that SNLW is ill-posed in the sense that the second order stochastic term is not a continuous function of time with values in spatial distributions. This shows that a standard method such as the Da Prato-Debussche trick or its variant, based on a higher order expansion, breaks down for α ≥ 1 2 . (ii) As for SNLH, we establish analogous results with a threshold given byThese examples show that in the case of rough noises, the existing well-posedness theory for singular stochastic PDEs breaks down before reaching the critical values (α = 3 4 in the wave case and α = 2 in the heat case) predicted by the scaling analysis (due to Deng, Nahmod, and Yue (2019) in the wave case and due to Hairer (2014) in the heat case).

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Section: Stochastic Nonlinear Heat Equationmentioning

confidence: 99%

Comparing the stochastic nonlinear wave and heat equations: a case study

Oh¹,

Okamoto

2021

Electron. J. Probab.

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“…As in the case of KdV discussed in Remark 1.1, we then have convergence of the sequence of regularized solutions for any regularization of the initial data in the appropriate Fourier-Lebesgue space when > 0. See also [25] for an analogous local well-posedness result in the context of the stochastic cubic NLS on T with almost space-time white noise. When = 0, however, the white noise defined in (1.1) almost surely belongs to F , (T) only for < − 1 , and thus the deterministic argument in [19,29,60] is no longer applicable to our problem.…”

Section: Statements Of the Well-posedness Resultsmentioning

confidence: 91%

“…We also point out that the case of the standard NLS (with the second-order dispersion) is out of reach at this point. See the introduction in [25] for a discussion on the criticality of this problem (in the context of the stochastic NLS with additive space-time white noise forcing).…”

Section: The = 0 Casementioning

confidence: 99%

Solving the 4NLS with white noise initial data

Tzvetkov

Wang

2020

Forum of Mathematics, Sigma

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We construct global-in-time singular dynamics for the (renormalized) cubic fourth-order nonlinear Schrödinger equation on the circle, having the white noise measure as an invariant measure. For this purpose, we introduce the ‘random-resonant / nonlinear decomposition’, which allows us to single out the singular component of the solution. Unlike the classical McKean, Bourgain, Da Prato-Debussche type argument, this singular component is nonlinear, consisting of arbitrarily high powers of the random initial data. We also employ a random gauge transform, leading to random Fourier restriction norm spaces. For this problem, a contraction argument does not work, and we instead establish the convergence of smooth approximating solutions by studying the partially iterated Duhamel formulation under the random gauge transform. We reduce the crucial nonlinear estimates to boundedness properties of certain random multilinear functionals of the white noise.

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“…27 For example, for the subcritical SQE on T 3 , the second order iterate (an analogue of z3 in (3.21)) gains one derivative as compared to the stochastic convolution. a recent work [38], where the second author (with Forlano and Y. Wang) established local well-posedness of (4.3) with a slightly smooth noise ∂ x −ε ξ, ε > 0.…”

Section: Remarks and Commentsmentioning

confidence: 99%