On the Parabolic and Hyperbolic Liouville Equations

We study the two-dimensional stochastic nonlinear heat equation (SNLH) and stochastic damped nonlinear wave equation (SdNLW) with an exponential nonlinearity λβe^βu, 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.