On the Parabolic and Hyperbolic Liouville Equations

Tadahiro Oh*, Tristan Robert, Yuzhao Wang

*Corresponding author for this work

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Abstract

We study the two-dimensional stochastic nonlinear heat equation (SNLH) and stochastic damped nonlinear wave equation (SdNLW) with an exponential nonlinearity λβeβu, forced by an additive space-time white noise. (i) We first study SNLH for general 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<β2<8π3+22≃1.37π. Our argument yields stability under the noise perturbation, thus improving Garban’s local well-posedness result (2020). (ii) In the defocusing case 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 < β2< 4 π. (iii) As for SdNLW in the defocusing case 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 Φ34-model) and prove local well-posedness of SdNLW for the range: 0<β2<32-1635π≃0.86π. 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 λ> 0 , these models formally preserve the associated Gibbs measures with the exponential nonlinearity. Under the same assumption on 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 λ∈ Rwithout using the positivity of the Gaussian multiplicative chaos. This proves local well-posedness of SNLH for the range 0<β2<43π≃1.33π, slightly smaller than that in (i), but provides Lipschitz continuity of the solution map in initial data as well as the noise.

Original languageEnglish
Pages (from-to)1281-1351
Number of pages71
JournalCommunications in Mathematical Physics
Volume387
Issue number3
Early online date28 Sept 2021
DOIs
Publication statusPublished - Nov 2021

Bibliographical note

Funding Information:
T.O. was supported by the European Research Council (grant no. 637995 “ProbDynDispEq” and grant no. 864138 “SingStochDispDyn"). T.R. was supported by the European Research Council (grant no. 637995 “ProbDynDispEq”). Y.W. was supported by supported by the EPSRC New Investigator Award (grant no. EP/V003178/1).

Publisher Copyright:
© 2021, The Author(s).

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Mathematical Physics

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