Dynamical Liouville

Garban, Christophe

doi:10.1016/j.jfa.2019.108351

“…Since we will be partially working in the parabolic setting, we will use the easy adaptation of their results to the parabolic scaling. This adaptation was previously stated and used in [18]. Here we only state the results that we use in this paper.…”

Section: A Criterion For Tightnessmentioning

confidence: 99%

Constructing a solution of the $(2+1)$-dimensional KPZ equation

Chatterjee¹,

Dunlap²

2020

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The (d + 1)-dimensional KPZ equation is the canonical model for the growth of rough d-dimensional random surfaces. A deep mathematical understanding of the KPZ equation for d = 1 has been achieved in recent years, and the case d ≥ 3 has also seen some progress. The most physically relevant case of d = 2, however, is not very wellunderstood mathematically, largely due to the renormalization that is required: in the language of renormalization group analysis, the d = 2 case is neither ultraviolet superrenormalizable like the d = 1 case nor infrared superrenormalizable like the d ≥ 3 case. Moreover, unlike in d = 1, the Cole-Hopf transform is not directly usable in d = 2 because solutions to the multiplicative stochastic heat equation are distributions rather than functions. In this article we show the existence of subsequential scaling limits as ε → 0 of Cole-Hopf solutions of the (2 + 1)dimensional KPZ equation with white noise mollified to spatial scale ε and nonlinearity multiplied by the vanishing factor | log ε| − 1 2 . We also show that the scaling limits obtained in this way do not coincide with solutions to the linearized equation, meaning that the nonlinearity has a non-vanishing effect. We thus propose our scaling limit as a notion of KPZ evolution in 2 + 1 dimensions.where ν, λ and D are strictly positive parameters andẆ denotes a standard space-time white noise on the two-dimensional torus T 2 = R 2 /Z 2 . More precisely, we define W to be a cylindrical Wiener process on L 2 (T 2 ) whose covariance operator is the identity, as in [13] or [3], and thenẆ is its (distributional) derivative in time. Thus, formally we have

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“…This provides another motivation to study well-posedness of the equations (1.1) and (1.2). We also point out that the exp( ) 2 -measure and the resulting Gaussian multiplicative chaos play an important role in Liouville quantum gravity [21,22,25,26,46,66]; see also a recent paper [30] for a nice exposition and further references therein. We also mention the works [1,4] on the elliptic exp( ) 2 -model.…”

Section: Parabolic and Hyperbolic Liouville Equationsmentioning

confidence: 97%

“…This requires us to introduce a proper renormalization, adapted to the exponential nonlinearity, to give a precise meaning to the equations. 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].…”

Section: Parabolic and Hyperbolic Liouville Equationsmentioning

confidence: 99%

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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“…In the next subsection, we provide precise statements of our main results. Before proceeding further, we mention the recent works [11,20,34] on the well-posedness theory for the stochastic heat and wave equations with an exponential nonlinearity in the two-dimensional setting.…”

Section: Dynamical Sine-gordon Modelmentioning

confidence: 99%

Invariant Gibbs dynamics for the dynamical sine-Gordon model

Oh¹,

Robert

²

,

Sosoe³

et al. 2020

Proceedings of the Royal Society of Edinburgh: Section A Mathem

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In this note, we study the hyperbolic stochastic damped sine-Gordon equation (SdSG), with a parameter β2 > 0, and its associated Gibbs dynamics on the two-dimensional torus. After introducing a suitable renormalization, we first construct the Gibbs measure in the range 0 < β2 < 4π via the variational approach due to Barashkov-Gubinelli (2018). We then prove almost sure global well-posedness and invariance of the Gibbs measure under the hyperbolic SdSG dynamics in the range 0 < β2 < 2π. Our construction of the Gibbs measure also yields almost sure global well-posedness and invariance of the Gibbs measure for the parabolic sine-Gordon model in the range 0 < β2 < 4π.

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Dynamical Liouville

Cited by 18 publications

References 48 publications

Constructing a solution of the $(2+1)$-dimensional KPZ equation

Constructing a solution of the $(2+1)$-dimensional KPZ equation

On the Parabolic and Hyperbolic Liouville Equations

Invariant Gibbs dynamics for the dynamical sine-Gordon model

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