Optimal integrability threshold for Gibbs measures associated with focusing NLS on the torus

Oh, Tadahiro; Sosoe, Philippe; Tolomeo, Leonardo

doi:10.1007/s00222-021-01080-y

Cited by 23 publications

(26 citation statements)

References 74 publications

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“…See Remark 5.10. We also mention related works [48,19,75,15,69,61,68] on the non-normalizability (and other issues) for focusing Gibbs measures.…”

mentioning

confidence: 99%

Focusing $Φ^4_3$-model with a Hartree-type nonlinearity

Oh¹,

Okamoto²,

Tolomeo³

2020

Preprint

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114

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We study a focusing Φ 4 3 -model with a Hartree-type nonlinearity, where the potential for the Hartree nonlinearity is given by the Bessel potential of order β. We first apply the variational argument introduced by Barashkov and Gubinelli (2018) and construct the focusing Hartree Φ 4 3 -measure for β > 2. We also show that the threshold β = 2 is sharp in the sense that the associated Gibbs measure is not normalizable for β < 2. Furthermore, we show that the following dichotomy holds at the critical value β = 2: normalizability in the weakly nonlinear regime and non-normalizability in the strongly nonlinear regime. We then establish a sharp almost sure global well-posedness result for the canonical stochastic quantization of the focusing Hartree Φ 4 3 -measure. Namely, we study the three-dimensional stochastic damped nonlinear wave equation (SdNLW) with a cubic nonlinearity of Hartree-type, forced by an additive space-time white noise. Using ideas from paracontrolled calculus, in particular from the recent work by Gubinelli, Koch, and the first author (2018), we prove local well-posedness of the focusing Hartree SdNLW for β > 2 (and β = 2 in the weakly nonlinear regime). In order to handle the resonant interaction, we rewrite the equation into a system of three unknowns. We then establish almost sure global well-posedness and invariance of the focusing Hartree Φ 4 3 -measure via Bourgain's invariant measure argument (1994, 1996). In view of the non-normalizability result, our almost sure global well-posedness result is sharp. In Appendix, we also consider the (parabolic) stochastic quantization for the focusing Hartree Φ 4 3 -measure and construct global-in-time invariant dynamics for β > 2 (and β = 2 in the weakly nonlinear regime).We also consider the Hartree Φ 4 3 -measure in the defocusing case. By adapting our argument from the focusing case, we first construct the defocusing Hartree Φ 4 3 -measure and the associated invariant dynamics for the defocusing Hartree SdNLW for β > 1. By introducing further renormalizations at β = 1 and β = 1 2 , we extend the construction of the defocusing Hartree Φ 4 3 -measure for β > 0, where the resulting measure is shown to be singular with respect to the reference Gaussian free field for 0 < β ≤ 1 2 .

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“…See Remark 5.10. We also mention related works [48,19,75,15,69,61,68] on the non-normalizability (and other issues) for focusing Gibbs measures.…”

mentioning

confidence: 99%

Focusing $Φ^4_3$-model with a Hartree-type nonlinearity

Oh¹,

Okamoto²,

Tolomeo³

2020

Preprint

Self Cite

114

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“…We point out that the argument in [31] shows non-normalizability only for large K ≫ 1 and thus we need to refine the argument to prove the divergence (1.17) for any K > 0. We also mention related works [27,11,42,10,36] on the non-normalizability (and other issues) for focusing Gibbs measures.…”

mentioning

confidence: 99%

A remark on Gibbs measures with log-correlated Gaussian fields

Oh¹,

Seong²,

Tolomeo³

2020

Preprint

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We study Gibbs measures with log-correlated base Gaussian fields on the ddimensional torus. In the defocusing case, the construction of such Gibbs measures follows from Nelson's argument. In this paper, we consider the focusing case with a quartic interaction. Using the variational formulation, we prove non-normalizability of the Gibbs measure. When d = 2, our argument provides an alternative proof of the non-normalizability result for the focusing Φ 4 2 -measure by Brydges and Slade (1996). We also go over the construction of the focusing Gibbs measure with a cubic interaction. In the appendices, we present (a) non-normalizability of the Gibbs measure for the two-dimensional Zakharov system and (b) the construction of focusing quartic Gibbs measures with smoother base Gaussian measures, showing a critical nature of the log-correlated Gibbs measure with a focusing quartic interaction.

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“…Our motivation is twofold. For one, we think that this case is of intrinsic interest -random initial data has been studied for a variety of PDEs, such as the nonlinear Schrödinger equation [3,4,14,22,25], the nonlinear wave equation [5,6], and the Navier-Stokes equations [24]. (This list of references is woefully incomplete.…”

Section: Introductionmentioning

confidence: 99%

The Yang-Mills heat flow with random distributional initial data

Cao¹,

Chatterjee²

2021

Preprint

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We construct local solutions to the Yang-Mills heat flow (in the DeTurck gauge) for a certain class of random distributional initial data, which includes the 3D Gaussian free field. The main idea, which goes back to work of Bourgain as well as work of Da Prato-Debussche, is to decompose the solution into a rougher linear part and a smoother nonlinear part, and to control the latter by probabilistic arguments. In a companion work, we use the main results of this paper to propose a way towards the construction of 3D Yang-Mills measures.

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Optimal integrability threshold for Gibbs measures associated with focusing NLS on the torus

Cited by 23 publications

References 74 publications

Focusing $Φ^4_3$-model with a Hartree-type nonlinearity

Focusing $Φ^4_3$-model with a Hartree-type nonlinearity

A remark on Gibbs measures with log-correlated Gaussian fields

The Yang-Mills heat flow with random distributional initial data

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