Quasi-invariant Gaussian measures for the cubic fourth order nonlinear Schrödinger equation in negative Sobolev spaces

Oh, Tadahiro; Seong, Kihoon

doi:10.48550/arxiv.2012.06732

Cited by 4 publications

(6 citation statements)

References 31 publications

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“…We present the theory via two notable examples, namely the Benjamin-Bona-Mahony (BBM) equation with dispersion β > 1 and the periodic quintic defocussing nonlinear Schrödinger equation (NLS), demonstrating that in this way we can improve on the previous analyses of [28] and [24] respectively. The result of [28] was extended to more involved models in [7,11,12,13,9,10,15,18,19,20,21,22,24,26]. We believe that, beyond the BBM and NLS equation, the idea of an exponential cut-off introduced in the present paper may be relevant in the context of some of these works and, more generally, in the study of quasi-invariant measures for Hamiltonian PDEs.…”

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confidence: 75%

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

Genovese¹,

Lucà²,

Tzvetkov³

2021

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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 α (T) for arbitrarily small α > 0 and invariance of the Gaussian measure associated with the H β/2 (T) norm.

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confidence: 75%

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

Genovese¹,

Lucà²,

Tzvetkov³

2021

Preprint

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“…This paper triggered a renewed interest in the subject from the viewpoint of dispersive PDEs, which translates into studying the evolution of random initial data (such as Brownian motion and related processes). For recent developments on the topic, see [2,6,7,8,9,10,11,12,14,19,20,21,22,23,24,26], although this list might be not exhaustive.…”

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confidence: 99%

Quasi-invariance of Gaussian measures for the periodic Benjamin-Ono-BBM equation

Genovese¹,

Lucà²,

Tzvetkov³

2021

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The BBM equation is a Hamiltonian PDE which revealed to be a very interesting test-model to study the transformation property of Gaussian measures along the flow, after [27].In this paper we study the BBM equation with critical dispersion (which is a Benjamin-Ono type model). We prove that the image of the Gaussian measures supported on fractional Sobolev spaces of increasing regularity are absolutely continuous, but we cannot identify the density, for which new ideas are needed.

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“…The earliest references we are aware of is [13], followed by [25,52,3,4]. Inspired by the work on invariant measures for the Benjamin-Ono equation [10,48,49,50], quasi-invariance of Gaussian measure for several dispersive models was obtained in recent years, see [9,14,16,11,12,18,31,34,35,36,37,39,43,47]. The method to identify the densities in Theorem 1.3 is inspired by recent works [9,15].…”

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confidence: 99%