Transport of Gaussian measures under the flow of one-dimensional fractional nonlinear Schrödinger equations

Forlano, Justin; Seong, Kihoon

doi:10.1080/03605302.2022.2053861

Cited by 6 publications

(3 citation statements)

References 47 publications

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“…Together with the result of the first author (and K. Seong) [30], that shows quasi invariance for (1.1) for every s > 1 2 , Theorem 1.1 extends quasi-invariance of the measure µ s to every s > s * (α).…”

supporting

confidence: 61%

“…Indeed, many results have appeared regarding the quasiinvariance of Gaussian measures for various different dispersive PDEs. In particular, there are results for quasi-invariance of the BBM and Benjamin-Ono equations [81,32,33], KdV type equations [72], wave equations [66,36,74], and Schrödinger equations [65,62,64,71,29,19,60,30,32]. The key underlying observation in this study is that the quasi-invariance of Gaussian measures is intimately tied to the dispersive character of the equation; see [62,74] for negative results for some dispersionless ODEs.…”

mentioning

confidence: 85%

“…where Q is an explicit function; see Lemma 5.1. Such a formula for the density has been known in the community for quite some time, however, it has been used in the context of quasi-invariance for dispersive PDEs only recently, initially by Debussche and Tsutsumi [19], and subsequently in [12,30,32]. Afterwards, by a bootstrap/Gronwall argument, we can derive an L p (µ s ) estimate for f t from an L p (µ s ) estimate for…”

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confidence: 99%

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Quasi-invariance of Gaussian measures of negative regularity for fractional nonlinear Schrödinger equations

Forlano¹,

Tolomeo²

2022

Preprint

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We consider the Cauchy problem for the fractional nonlinear Schrödinger equation (FNLS) on the one-dimensional torus with cubic nonlinearity and high dispersion parameter α > 1, subject to a Gaussian random initial data of negative Sobolev regularity σ < s − 1 2 , for s ≤ 1 2 . We show that for all s * (α) < s ≤ 1 2 , the equation is almost surely globally well-posed. Moreover, the associated Gaussian measure supported on H s (T) is quasi-invariant under the flow of the equation. For α < 1 20 (17 + 3 √ 21) ≈ 1.537, the regularity of the initial data is lower than the one provided by the deterministic wellposedness theory.We obtain this result by following the approach of DiPerna-Lions (1989); first showing global-in-time bounds for the solution of the infinite-dimensional Liouville equation for the transport of the Gaussian measure, and then transferring these bounds to the solution of the equation by adapting Bourgain's invariant measure argument to the quasi-invariance setting. This allows us to bootstrap almost sure global bounds for the solution of (FNLS) from its probabilistic local well-posedness theory.

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confidence: 61%

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confidence: 85%

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confidence: 99%

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Quasi-invariance of Gaussian measures of negative regularity for fractional nonlinear Schrödinger equations

Forlano¹,

Tolomeo²

2022

Preprint

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Blow-Up for the 1D Cubic NLS

Banica,

Lucà,

Tzvetkov

et al. 2024

Commun. Math. Phys.

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Transport of Gaussian measures with exponential cut-off for Hamiltonian PDEs

Genovese

Lucà

Tzvetkov

2023

JAMA

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We show that introducing an exponential cut-off on a suitable Sobolev norm facilitates the proof of quasi-invariance of Gaussian measures with respect to Hamiltonian PDE flows and allows us to establish the exact Jacobi formula for the density. We exploit this idea in two different contexts, namely the periodic fractional Benjamin–Bona–Mahony (BBM) equation with dispersion β > 1 and the periodic one-dimensional quintic defocussing nonlinear Schrödinger equation (NLS). For the BBM equation we study the transport of the cut-off Gaussian measures on fractional Sobolev spaces, while for the NLS equation we study the measures based on the modified energies introduced by Planchon–Visciglia and the third author. Moreover, for the BBM equation we also show almost sure global well-posedness for data in $${C^\alpha }(\mathbb{T})$$ C α ( T ) for arbitrarily small α > 0 and invariance of the Gaussian measure associated with the $${H^{\beta /2}}(\mathbb{T})$$ H β / 2 ( T ) norm.

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Transport of Gaussian measures under the flow of one-dimensional fractional nonlinear Schrödinger equations

Cited by 6 publications

References 47 publications

Quasi-invariance of Gaussian measures of negative regularity for fractional nonlinear Schrödinger equations

Quasi-invariance of Gaussian measures of negative regularity for fractional nonlinear Schrödinger equations

Blow-Up for the 1D Cubic NLS

Transport of Gaussian measures with exponential cut-off for Hamiltonian PDEs

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