A pedestrian approach to the invariant Gibbs measures for the 2-d defocusing nonlinear Schrödinger equations

Oh, Tadahiro; Thomann, Laurent

doi:10.1007/s40072-018-0112-2

Cited by 51 publications

(124 citation statements)

References 40 publications

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“…With this general background in mind, let us now go back to the consideration of equation (1). Our approach to the model will directly follow a series of investigations [3,4,12,19,23] devoted to the study of stochastic wave (or Schrödinger) equations involving a polynomial drift term. Our study can more specifically be seen as a fractional extension of the results of [12] for the white-noise situation.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…In the latter situation, and when turning to the study of the auxiliary equation (3), one must then cope with the problem of interpreting the product Ψ 2 . Just as in [12,19], we will actually understand this product in the Wick sense, which, again, can be made rigorous through an approximation/renormalization procedure:…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…Thus, our results should morally be read as local results, both in time and in space, for the real "target" equation, that is the equation with ρ ≡ 1. What refrained us to formulate the problem on a torus (just as in [12,19]) is the consideration of the fractional noise, which is more convenient to define and handle on the whole Euclidean space.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

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A nonlinear wave equation with fractional perturbation

Deya¹

2019

Ann. Probab.

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We study a d-dimensional wave equation model (2 ≤ d ≤ 4) with quadratic non-linearity and stochastic forcing given by a space-time fractional noise. Two different regimes are exhibited, depending on the Hurst parameter H = (H 0 , . . . ,, the model must be interpreted in the Wick sense, through a renormalization procedure.Our arguments essentially rely on a fractional extension of the considerations of [12] for the twodimensional white-noise situation, and more generally follow a series of investigations related to stochastic wave models with polynomial perturbation.Since the pioneering works of Mandelbrot and Van Ness, fractional noises have been considered as very natural stochastic perturbation models, that offer more flexibility than classical white-noise-driven equations. The involvement of fractional inputs first occured in the setting of standard differential equations and, even in this simple context, is known to raise numerous difficulties due to the nonmartingale nature of the process. Sophisticated alternatives to Ito theory must then come into the picture, whether fractional calculus, Malliavin calculus or rough paths theory, to mention just the most standard methods.More recently, fractional (multiparameter) noises have also appeared within SPDE models. A first widely-used example is given by white-in-time colored-in-space Gaussian noises, that can be treated in the classical framework of Walsh's martingale-measure theory [25], or with Da Prato-Zabczyk's infinitedimensional approach to stochastic calculus [6]. Such noise models have thus been applied to a large class

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Section: Introduction and Main Resultsmentioning

confidence: 99%

Section: Introduction and Main Resultsmentioning

confidence: 99%

Section: Introduction and Main Resultsmentioning

confidence: 99%

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A nonlinear wave equation with fractional perturbation

Deya¹

2019

Ann. Probab.

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“…A direct computation shows that the gauged function, which we still denote by u, satisfies the following renormalized 4NLS: This renormalization appears as an equivalent formulation of the Wick renormalization in Euclidean quantum field theory [4,34,35]. (By viewing u as a complex-valued Gaussian random variables, the Wick renormalization of |u| 2 u is nothing but a projection onto the Wiener homogeneous chaoses of order three.)…”

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confidence: 99%

Global Well-Posedness of the Periodic Cubic Fourth Order NLS in Negative Sobolev Spaces

Wang

2018

Forum of Mathematics, Sigma

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We consider the Cauchy problem for the cubic fourth order nonlinear Schrödinger equation (4NLS) on the circle. In particular, we prove global well-posedness of the renormalized 4NLS in negative Sobolev spaces H s (T), s > − , by establishing an energy estimate for the difference of two solutions with the same initial condition. For this purpose, we perform an infinite iteration of normal form reductions on the H s -energy functional, allowing us to introduce an infinite sequence of correction terms to the H s -energy functional in the spirit of the I -method. In fact, the main novelty of this paper is this reduction of the H s -energy functionals (for a single solution and for the difference of two solutions with the same initial condition) to sums of infinite series of multilinear terms of increasing degrees.

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“…Skorokhod's theorem then allowed them to construct a function u as an almost sure limit of the solutions u N to the truncated dynamics (distributed according to ν N ), yielding almost sure global existence. See [70,32] for more on this method. This construction of global-in-time solutions is closely related to the notion of martingale solutions in the field of stochastic PDEs.…”

Section: 2mentioning

confidence: 99%