Invariant Gibbs measures for the three dimensional cubic nonlinear wave equation

Bringmann, Bjoern; Yu, Dao‐Yi; Nahmod, Andrea R.; Yue, Haitian

doi:10.48550/arxiv.2205.03893

Cited by 7 publications

(10 citation statements)

References 58 publications

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“…where the noise ζ is primarily taken to be a space-time white noise ξ. Here, N (u) denotes a nonlinearity which may be of a power-type [34,35,36,65,57,80,58,13,59,70,15] and trigonometric and exponential nonlinearities [63,66,64]. We also mention the works [69,61,60,76,71] on the (deterministic) nonlinear wave equations (1.4) with rough random initial data and [24,25,56] on SNLW with more singular (both in space and time) noises.…”

Section: As For the Construction Of The Limiting φ K+1mentioning

confidence: 99%

Hyperbolic $P(Φ)_2$-model on the plane

Oh¹,

Tolomeo²,

Wang³

et al. 2022

Preprint

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We study the hyperbolic Φ k+1 2 -model on the plane. By establishing coming down from infinity for the associated stochastic nonlinear heat equation (SNLH) on the plane, we first construct a Φ k+1 2 -measure on the plane as a limit of the Φ k+1 2 -measures on large tori. We then study the canonical stochastic quantization of the Φ k+1 2 -measure on the plane thus constructed, namely, we study the defocusing stochastic damped nonlinear wave equation forced by an additive space-time white noise (= the hyperbolic Φ k+1 2 -model) on the plane. In particular, by taking a limit of the invariant Gibbs dynamics on large tori constructed by the first two authors with Gubinelli and Koch (2021), we construct invariant Gibbs dynamics for the hyperbolic Φ k+1 2 -model on the plane. Our main strategy is to develop further the ideas from a recent work on the hyperbolic Φ 3 3 -model on the three-dimensional torus by the first two authors and Okamoto ( 2021), and to study convergence of the socalled enhanced Gibbs measures, for which coming down from infinity for the associated SNLH with positive regularity plays a crucial role. By combining wave and heat analysis together with ideas from optimal transport theory, we then conclude global well-posedness of the hyperbolic Φ k+1 2 -model on the plane and invariance of the associated Gibbs measure. As a byproduct of our argument, we also obtain invariance of the limiting Φ k+1 2 -measure on the plane under the dynamics of the parabolic Φ k+1 2 -model.

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Section: As For the Construction Of The Limiting φ K+1mentioning

confidence: 99%

Hyperbolic $P(Φ)_2$-model on the plane

Oh¹,

Tolomeo²,

Wang³

et al. 2022

Preprint

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show abstract

“…In three dimensions, this is closely related to the famous Gibbs measure invariance problem for cubic NLS (i.e. invariance of the Φ 4 3 measure under the Schrödinger dynamics), which is the only Gibbs measure invariance problem that still remains open in all polynomial settings after the works of [2,3,45,41,19,5]. In addition, energy cascade behavior for NLS can also be observed at the level of (WKE) [39,27], and proving such cascade dynamics for the NLS equation on the unit torus is a problem of great interest [4].…”

Section: Introduction and The Physical Theorymentioning

confidence: 98%

Rigorous justification of the wave kinetic theory

Yu¹,

Hani²

2022

Preprint

Self Cite

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The main purpose of this expository note is to give a short account of the recent developments in mathematical wave kinetic theory. After reviewing the physical theory, we explain the importance of the notion of a "scaling law", which dictates the relation between the asymptotic parameters as the kinetic limit is taken. This sets some natural limitations on the kinetic approximation that were not precisely understood in the literature as far as we know. We then describe our recent and upcoming works that give the first full, mathematically rigorous, derivation of the wave kinetic theory at the natural kinetic timescale. The key new ingredient is a delicate analysis of the diagrammatic expansion that allows to a) uncover highly elaborate cancellations at arbitrary large order of diagrams, and b) overcome difficulties coming from factorial divergences in the expansion and the criticality of the problem. The results mentioned in this note appear in our recent works [16,17] as well as an upcoming one in [18].

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“…We conclude this paragraph by mentioning that the paracontrolled approach was further developed in a spectacular way in a series of papers of Deng, Nahmod and Yue [14,15] and Sun, Tzvetkov [36] bringing tools from random matrix theory and introducing powerful methods such as the random averaging operators and the random tensors. More recently, the paracontrolled approach has taken a step forward, and were successfully implied in the resolution by Bringmann, Deng, Nahmod and Yue [7] of the φ 3 4 problem for NLW.…”

Section: Introductionmentioning

confidence: 99%

Refined probabilistic local well-posedness for a cubic Schrödinger half-wave equation

Camps¹,

Gassot²,

Ibrahim³

2022

Preprint

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We obtain probabilistic local well-posedness in quasilinear regimes for the Schrödinger half-wave equation with a cubic nonlinearity. We need to use a refined ansatz because of the lack of probabilistic smoothing in the Picard's iterations, which is due to the high-low-low nonlinear interactions. The proof is an adaptation of the method of Bringmann on the derivative nonlinear wave equation [6] to Schrödinger-type equations. In addition, we discuss ill-posedness results for this equation.

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Invariant Gibbs measures for the three dimensional cubic nonlinear wave equation

Cited by 7 publications

References 58 publications

Hyperbolic $P(Φ)_2$-model on the plane

Hyperbolic $P(Φ)_2$-model on the plane

Rigorous justification of the wave kinetic theory

Refined probabilistic local well-posedness for a cubic Schrödinger half-wave equation

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