On the parabolic and hyperbolic Liouville equations

TL;DR

This paper establishes local and global well-posedness and Gibbs measure invariance for two-dimensional stochastic nonlinear heat and wave equations with exponential nonlinearities, depending on parameters, advancing understanding of these complex stochastic PDEs.

Contribution

It provides the first rigorous analysis of the stochastic parabolic and hyperbolic Liouville equations with exponential nonlinearities, including well-posedness and measure invariance results.

Findings

Proved local and global well-posedness depending on parameters.

Established invariance of Gibbs measures for the equations.

Analyzed the influence of the sign of λ and size of β on solutions.

Abstract

We study the two-dimensional stochastic nonlinear heat equation (SNLH) and stochastic damped nonlinear wave equation (SdNLW) with an exponential nonlinearity $\lambda\beta e^{\beta u }$, forced by an additive space-time white noise. We prove local and global well-posedness of these equations, depending on the sign of $\lambda$ and the size of $\beta^2 > 0$, and invariance of the associated Gibbs measures. See the abstract of the paper for a more precise abstract. (Due to the limit on the number of characters for an abstract set by arXiv, the full abstract can not be displayed here.)