Random tensors, propagation of randomness, and nonlinear dispersive equations

Deng, Yu; Nahmod, Andrea R.; Yue, Haitian

doi:10.48550/arxiv.2006.09285

Cited by 14 publications

(38 citation statements)

References 55 publications

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“…This outline is rather simplistic; in reality there are other almost-leading diagrams whose contributions have to be analyzed separately. Moreover, the problem of estimating the diagrams is probabilistically critical in the sense of [10], which is added to the factorial growth of the number of diagrams, to make the execution of this outline far from trivial. We shall review some elements of that proof in Section 3 below, and also refer the reader to Section 3 of [9] for a more detailed exposition.…”

Section: Introductionmentioning

confidence: 99%

Propagation of chaos and the higher order statistics in the wave kinetic theory

Deng¹,

Hani²

2021

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This manuscript continues and extends in various directions the result in [9], which gave a full derivation of the wave kinetic equation (WKE) from the nonlinear Schrödinger (NLS) equation in dimensions d ≥ 3. The wave kinetic equation describes the effective dynamics of the second moments of the Fourier modes of the NLS solution at the kinetic timescale, and in the kinetic limit in which the size of the system diverges to infinity and the strength of the nonlinearity vanishes asymptotically according to a specified scaling law.Here, we investigate the behavior of the joint distribution of these Fourier modes and derive their effective limit dynamics at the kinetic timescale. In particular, we prove propagation of chaos in the wave setting: initially independent Fourier modes retain this independence in the kinetic limit. Such statements are central to the formal derivations of all kinetic theories, dating back to the work of Boltzmann (Stosszahlansatz). We obtain this by deriving the asymptotics of the higher Fourier moments, which are shown to be given by solutions of the so-called wave kinetic heirarchy (WKH) with factorized initial data, hence giving a rigorous justification of the latter system. We treat both Gaussian and non-Gaussian initial distributions. In the Gaussian setting, we prove propagation of Gaussianity as we show that the asymptotic distribution retains the Gaussianity of the initial data in the limit. In the non-Gaussian setting, we derive the limiting equations for the higher order moments, as well as for the density function (PDF) of the solution. Some of the results we prove were conjectured in the physics literature, others appear to be new. This gives a complete description of the statistics of the solutions in the kinetic limit.

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Section: Introductionmentioning

confidence: 99%

Propagation of chaos and the higher order statistics in the wave kinetic theory

Deng¹,

Hani²

2021

Preprint

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“…It might be possible to refine the argument below for the multilinear estimates involved in the proof of Theorem 1.5 in the case of High×High→High interactions to show that we can take α > 3 2 in Theorem 1.5, and that this threshold for the dispersion solely comes from the control on High×Low→High interactions with a linear evolution of the random initial data as a high regularity input. This is precisely the bad interaction that is amenable to the more sophisticated method of random averaging operators and random tensors developed recently in [26,27]. Thus we believe that there is a chance that Theorem 1.5 can be improved all the way down to the range α > 1.…”

Section: Remark 14mentioning

confidence: 94%

“…See also the discussions in a similar context in [57,58]. However, note that in the critical case α = d, the exponential NLS appears to be probabilistic critical in the sense of [26,27], and it is not clear at all that the methods developed in these papers could handle this case due to combinatorics losses. Namely, after decomposing the (renormalized) nonlinearity in (1.17) as…”

Section: Remark 14mentioning

confidence: 99%

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Invariant Gibbs measure for a Schrodinger equation with exponential nonlinearity

Robert¹

2021

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We investigate the invariance of the Gibbs measure for the fractional Schrödinger equation of exponential type (expNLS) i∂tu + (−∆) α 2 u = 2γβe β|u| 2 u on ddimensional compact Riemannian manifolds M d , for a dispersion parameter α > d, some coupling constant β > 0, and γ = 0. (i) We first study the construction of the Gibbs measure for (expNLS). We prove that in the defocusing case γ > 0, the measure is well-defined in the whole regime α > d and β > 0 (Theorem 1.1 (i)), while in the focusing case γ < 0 its partition function is always infinite for any α > d and β > 0, even with a mass cut-off of arbitrary small size (Theorem 1.1 (ii)). (ii) We then study the dynamics (expNLS) with random initial data of low regularity. We first use a compactness argument to prove weak invariance of the Gibbs measure in the whole regime α > d and 0 < β < β ⋆ α for some natural parameter 0In the large dispersion regime α > 2d, we can improve this result by constructing a local deterministic flow for (expNLS) for any β > 0. Using the Gibbs measure, we prove that solutions are almost surely global for 0 < β ≪ β ⋆ α , and that the Gibbs measure is invariant (Theorem 1.3 (ii)). (iii) Finally, in the particular case d = 1 and M = T, we are able to exploit some probabilistic multilinear smoothing effects to build a probabilistic flow for (expNLS) for 1 + √ 2 2 < α ≤ 2, locally for arbitrary β > 0 and globally for 0 < β ≪ β ⋆ α (Theorem 1.5).2020 Mathematics Subject Classification. 35Q41. Key words and phrases. dispersive equation; Schrödinger equation; Gibbs measure. 1 2 T. ROBERT 4.1. Local well-posedness for high dispersion 30 4.2. Almost sure global well-posedness and invariance of the Gibbs measure 31 5. Strong well-posedness on the circle 35 5.1. Function spaces and large deviation bounds 35 5.2. The gauge transform 39 5.3. A local well-posedness result 41 5.4. Proof of the multilinear estimates 45 5.5. Almost sure global well-posedness and invariance of the Gibbs measure 52 References 58

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“…In comparison with Bourgain's trick (1.8), the advantage of the random expansion (1.10) is that the nonlinear remainder z lives at regularity 1´, which is above the deterministic threshold for local well-posedness. More recently, the random averaging operators were generalized to random tensors in [DNY20].…”

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confidence: 99%