Invariant Gibbs dynamics for the dynamical sine-Gordon model

Oh, Tadahiro; Robert, Tristan; Sosoe, Philippe; Wang, Yuzhao

doi:10.1017/prm.2020.68

“…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%

“…In [29], by introducing an appropriate time-dependent renormalization, the authors proved local well-posedness of (a renormalized version of) (1.6) on T 2 . See [30,31,48,41,46,49,47] for further work on SNLW with singular stochastic forcing. We also mention the work [21,22] by Deya on SNLW with more singular (both in space and time) noises on bounded domains in R d and the work [55] on global well-posedness of the cubic SNLW on R 2 .…”

Section: Stochastic Nonlinear Wave Equationmentioning

confidence: 99%

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

Oh¹,

Okamoto

²

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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“…In recent years, we have seen a tremendous development in the study of singular stochastic PDEs, in particular in the parabolic setting [17,19,20,30,34,38,39,42,49,55]. 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 [15,23,24,[35][36][37][58][59][60][61][62][63][64][65]69,77]. On the two-dimensional torus T 2 , the stochastic heat and wave equations with a monomial nonlinearity u k (see (1.3) and (1.4) below) have been studied in [20,35,37].…”

Section: Parabolic and Hyperbolic Liouville Equationsmentioning

confidence: 99%

“…In the two-dimensional case, the stochastic convolution is only a distribution, making the problem much more delicate. To illustrate this, we first discuss the case of the sine-Gordon models on T 2 studied in [19,42,63,64]. In the parabolic setting, Hairer-Shen [42] and Chandra-Hairer-Shen [19] studied the following parabolic sine-Gordon model on T 2 : ∂ t u + 1 2 (1 − )u + sin(βu) = ξ.…”

Section: Parabolic and Hyperbolic Liouville Equationsmentioning

confidence: 99%

“…By using the theory of regularity structures [39], Chandra, Hairer, and Shen proved local well-posedness of (1.5) for the entire subcritical regime 0 < β 2 < 8π . More recently, the authors with P. Sosoe studied the hyperbolic counterpart of the sine-Gordon problem [63,64]. Due to a weaker smoothing property of the wave propagator, however, the resulting solution theory is much less satisfactory than that in the parabolic case; in the damped wave case, local well-posedness was established only for 0 < β 2 < 2π .…”

Section: Parabolic and Hyperbolic Liouville Equationsmentioning

confidence: 99%

See 1 more Smart Citation

On the Parabolic and Hyperbolic Liouville Equations

Oh

¹

,

Robert

²

,

Wang

³

2021

Commun. Math. Phys.

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We study the two-dimensional stochastic nonlinear heat equation (SNLH) and stochastic damped nonlinear wave equation (SdNLW) with an exponential nonlinearity $$\lambda \beta e^{\beta u }$$ λ β e β u , forced by an additive space-time white noise. (i) We first study SNLH for general $$\lambda \in {\mathbb {R}}$$ λ ∈ R . By establishing higher moment bounds of the relevant Gaussian multiplicative chaos and exploiting the positivity of the Gaussian multiplicative chaos, we prove local well-posedness of SNLH for the range $$0< \beta ^2 < \frac{8 \pi }{3 + 2 \sqrt{2}} \simeq 1.37 \pi $$ 0 < β 2 < 8 π 3 + 2 2 ≃ 1.37 π . Our argument yields stability under the noise perturbation, thus improving Garban’s local well-posedness result (2020). (ii) In the defocusing case $$\lambda >0$$ λ > 0 , we exploit a certain sign-definite structure in the equation and the positivity of the Gaussian multiplicative chaos. This allows us to prove global well-posedness of SNLH for the range: $$0< \beta ^2 < 4\pi $$ 0 < β 2 < 4 π . (iii) As for SdNLW in the defocusing case $$\lambda > 0$$ λ > 0 , we go beyond the Da Prato-Debussche argument and introduce a decomposition of the nonlinear component, allowing us to recover a sign-definite structure for a rough part of the unknown, while the other part enjoys a stronger smoothing property. As a result, we reduce SdNLW into a system of equations (as in the paracontrolled approach for the dynamical $$\Phi ^4_3$$ Φ 3 4 -model) and prove local well-posedness of SdNLW for the range: $$0< \beta ^2 < \frac{32 - 16\sqrt{3}}{5}\pi \simeq 0.86\pi $$ 0 < β 2 < 32 - 16 3 5 π ≃ 0.86 π . This result (translated to the context of random data well-posedness for the deterministic nonlinear wave equation with an exponential nonlinearity) solves an open question posed by Sun and Tzvetkov (2020). (iv) When $$\lambda > 0$$ λ > 0 , these models formally preserve the associated Gibbs measures with the exponential nonlinearity. Under the same assumption on $$\beta $$ β as in (ii) and (iii) above, we prove almost sure global well-posedness (in particular for SdNLW) and invariance of the Gibbs measures in both the parabolic and hyperbolic settings. (v) In Appendix, we present an argument for proving local well-posedness of SNLH for general $$\lambda \in {\mathbb {R}}$$ λ ∈ R without using the positivity of the Gaussian multiplicative chaos. This proves local well-posedness of SNLH for the range $$0< \beta ^2 < \frac{4}{3} \pi \simeq 1.33 \pi $$ 0 < β 2 < 4 3 π ≃ 1.33 π , slightly smaller than that in (i), but provides Lipschitz continuity of the solution map in initial data as well as the noise.

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On the two-dimensional hyperbolic stochastic sine-Gordon equation

Oh

¹

,

Robert

²

,

Sosoe

³

et al. 2020

Stoch PDE: Anal Comp

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We study the two-dimensional stochastic sine-Gordon equation (SSG) in the hyperbolic setting. In particular, by introducing a suitable time-dependent renormalization for the relevant imaginary multiplicative Gaussian chaos, we prove local wellposedness of SSG for any value of a parameter β 2 > 0 in the nonlinearity. This exhibits sharp contrast with the parabolic case studied by Hairer and Shen (2016) and Chandra, Hairer, and Shen (2018), where the parameter is restricted to the subcritical range: 0 < β 2 < 8π. We also present a triviality result for the unrenormalized SSG.2010 Mathematics Subject Classification. 35L71, 60H15.

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Invariant Gibbs dynamics for the dynamical sine-Gordon model

Cited by 28 publications

References 28 publications

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

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

On the Parabolic and Hyperbolic Liouville Equations

On the two-dimensional hyperbolic stochastic sine-Gordon equation

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