On the Parabolic and Hyperbolic Liouville Equations

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