Invariant Gibbs measures for the 2-<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>d</mml:mi></mml:math> defocusing nonlinear wave equations

Oh, Tadahiro; Thomann, Laurent

doi:10.5802/afst.1620

Cited by 50 publications

(74 citation statements)

References 31 publications

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“…Therefore, also π( t (u; ξ)) is a function of π(u), and moreover π( t (u; ξ)) = L(t)π(u), (38) so the projections of the flows for ( 37) and (4) coincide.…”

Section: Txmentioning

confidence: 97%

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Unique Ergodicity for a Class of Stochastic Hyperbolic Equations with Additive Space-Time White Noise

Tolomeo

2020

Commun. Math. Phys.

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In this paper, we consider a certain class of second order nonlinear PDEs with damping and space-time white noise forcing, posed on the d-dimensional torus. This class includes the wave equation for $$d=1$$ d = 1 and the beam equation for $$d\le 3$$ d ≤ 3 . We show that the Gibbs measure is the unique invariant measure for this system. Since the flow does not satisfy the strong Feller property, we introduce a new technique for showing unique ergodicity. This approach may be also useful in situations in which finite-time blowup is possible.

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“…Therefore, also π( t (u; ξ)) is a function of π(u), and moreover π( t (u; ξ)) = L(t)π(u), (38) so the projections of the flows for ( 37) and (4) coincide.…”

Section: Txmentioning

confidence: 97%

“…Following ideas first appearing in Bourgain's seminal paper [1] and in the works of McKean-Vasinski [33] and McKean [34,35], there have been many developments in proving invariance of the Gibbs measure for deterministic ispersive PDEs (see for instance [2][3][4][5][6][7]9,24,[38][39][40]).…”

Section: Introductionmentioning

confidence: 99%

Unique Ergodicity for a Class of Stochastic Hyperbolic Equations with Additive Space-Time White Noise

Tolomeo

2020

Commun. Math. Phys.

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“…We now proceed with the construction and renormalization of several stochastic objects. Similar constructions are standard in the probability theory literature and a comprehensive and well-written introduction can be found in [23,30,36]. In order to make this section accessible to readers with a primary background in dispersive PDEs, however, we include full details.…”

Section: Stochastic Objects and Renormalizationmentioning

confidence: 99%

“…After this renormalization, one can show (cf. [36]) that the densities d 4 2;N /dg 2 converge in L q (g 2 ) for all 1 ≤ q < ∞ and we can define 4 2 as the limit (in totalvariation) of 4 2;N as N → ∞. As in one spatial dimension, the 4 2 -model is absolutely continuous with respect to the Gaussian free field g 2 .…”

Section: Introductionmentioning

confidence: 96%

Invariant Gibbs measures for the three-dimensional wave equation with a Hartree nonlinearity I: measures

Bringmann

2021

Stoch PDE: Anal Comp

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In this two-paper series, we prove the invariance of the Gibbs measure for a three-dimensional wave equation with a Hartree nonlinearity. The main novelty is the singularity of the Gibbs measure with respect to the Gaussian free field. The singularity has several consequences in both measure-theoretic and dynamical aspects of our argument. In this paper, we construct and study the Gibbs measure. Our approach is based on earlier work of Barashkov and Gubinelli for the $$\Phi ^4_3$$ Φ 3 4 -model. Most importantly, our truncated Gibbs measures are tailored towards the dynamical aspects in the second part of the series. In addition, we develop new tools dealing with the non-locality of the Hartree interaction. We also determine the exact threshold between singularity and absolute continuity of the Gibbs measure depending on the regularity of the interaction potential.

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“…Over the last decade, we have seen a tremendous development in the study of singular stochastic PDEs, in particular in the parabolic setting [32,33,28,10,36,39,12,11,8,9]. Over the last few years, we have also witnessed a rapid progress in the theoretical understanding of nonlinear wave equations with singular stochastic forcing and/or rough random initial data [51,29,30,31,44,48,41,43,46,49,47,42,7]. While the regularity theory in the parabolic setting is well understood, the understanding of the solution theory in the hyperbolic/dispersive setting has been rather poor.…”

Section: Singular Stochastic Pdesmentioning

confidence: 99%

Comparing the stochastic nonlinear wave and heat equations: a case study

Oh¹,

Okamoto

2021

Electron. J. Probab.

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We study the two-dimensional stochastic nonlinear wave equation (SNLW) and stochastic nonlinear heat equation (SNLH) with a quadratic nonlinearity, forced by a fractional derivative (of order α > 0) of a space-time white noise. In particular, we show that the well-posedness theory breaks at α = 1 2 for SNLW and at α = 1 for SNLH. This provides a first example showing that SNLW behaves less favorably than SNLH. (i) As for SNLW, Deya (2020) essentially proved its local well-posedness for 0 < α < 1 2 . We first revisit this argument and establish multilinear smoothing of order 1 4 on the second order stochastic term in the spirit of a recent work by Gubinelli, Koch, and Oh (2018). This allows us to simplify the local well-posedness argument for some range of α. On the other hand, when α ≥ 1 2 , we show that SNLW is ill-posed in the sense that the second order stochastic term is not a continuous function of time with values in spatial distributions. This shows that a standard method such as the Da Prato-Debussche trick or its variant, based on a higher order expansion, breaks down for α ≥ 1 2 . (ii) As for SNLH, we establish analogous results with a threshold given byThese examples show that in the case of rough noises, the existing well-posedness theory for singular stochastic PDEs breaks down before reaching the critical values (α = 3 4 in the wave case and α = 2 in the heat case) predicted by the scaling analysis (due to Deng, Nahmod, and Yue (2019) in the wave case and due to Hairer (2014) in the heat case).

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Invariant Gibbs measures for the 2-d defocusing nonlinear wave equations

Cited by 50 publications

References 31 publications

Unique Ergodicity for a Class of Stochastic Hyperbolic Equations with Additive Space-Time White Noise

Unique Ergodicity for a Class of Stochastic Hyperbolic Equations with Additive Space-Time White Noise

Invariant Gibbs measures for the three-dimensional wave equation with a Hartree nonlinearity I: measures

Comparing the stochastic nonlinear wave and heat equations: a case study

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