On the Parabolic and Hyperbolic Liouville Equations

Abstract

We study the two-dimensional stochastic nonlinear heat equation (SNLH) and stochastic damped nonlinear wave equation (SdNLW) with an exponential nonlinearity $$\\lambda \\beta e^{\\beta u }$$\n \n λ\n β\n \n e\n \n β\n u\n \n \n \n , forced by an additive space-time white noise. (i)\xa0We first study SNLH for general $$\\lambda \\in {\\mathbb {R}}$$\n \n λ\n ∈\n R\n \n . 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 $$0< \\beta ^2 < \\frac{8 \\pi }{3 + 2 \\sqrt{2}} \\simeq 1.37 \\pi $$\n \n 0\n <\n \n β\n 2\n \n <\n \n \n 8\n π\n \n \n 3\n +\n 2\n \n 2\n \n \n \n ≃\n 1.37\n π\n \n . Our argument yields stability under the noise perturbation, thus improving Garban’s local well-posedness result (2020). (ii)\xa0In the defocusing case $$\\lambda >0$$\n \n λ\n >\n 0\n \n , 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: $$0< \\beta ^2 < 4\\pi $$\n \n 0\n <\n \n β\n 2\n \n <\n 4\n π\n \n . (iii) As for SdNLW in the defocusing case $$\\lambda > 0$$\n \n λ\n >\n 0\n \n , 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 $$\\Phi ^4_3$$\n \n Φ\n 3\n 4\n \n -model) and prove local well-posedness of SdNLW for the range: $$0< \\beta ^2 < \\frac{32 - 16\\sqrt{3}}{5}\\pi \\simeq 0.86\\pi $$\n \n 0\n <\n \n β\n 2\n \n <\n \n \n 32\n -\n 16\n \n 3\n \n \n 5\n \n π\n ≃\n 0.86\n π\n \n . 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)\xa0When $$\\lambda > 0$$\n \n λ\n >\n 0\n \n , these models formally preserve the associated Gibbs measures with the exponential nonlinearity. Under the same assumption on $$\\beta $$\n β\n 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 $$\\lambda \\in {\\mathbb {R}}$$\n \n λ\n ∈\n R\n \n without using the positivity of the Gaussian multiplicative chaos. This proves local well-posedness of SNLH for the range $$0< \\beta ^2 < \\frac{4}{3} \\pi \\simeq 1.33 \\pi $$\n \n 0\n <\n \n β\n 2\n \n <\n \n 4\n 3\n \n π\n ≃\n 1.33\n π\n \n , slightly smaller than that in (i), but provides Lipschitz continuity of the solution map in initial data as well as the noise.