Invariant Gibbs dynamics for the hyperbolic sinh-Gordon model

TL;DR

This paper proves global well-posedness and invariance of the Gibbs measure for the hyperbolic sinh-Gordon model on a 2D torus, introducing a novel $L^ Infty$-based approach that advances understanding of exponential nonlinear wave equations.

Contribution

It develops a new $L^ Infty$-based well-posedness framework for exponential nonlinear wave equations, enabling analysis of the sinh-Gordon model's Gibbs dynamics and invariance.

Findings

Established global well-posedness for certain parameter ranges.

Proved invariance of the Gibbs measure under the flow.

Improved well-posedness results for the hyperbolic Liouville model.

Abstract

We study the hyperbolic defocusing sinh-Gordon model with parameter $\beta^2>0$ and its associated Gibbs dynamics on the two-dimensional torus. We establish global well-posedness of the model for a certain range of parameters $\beta^2>0$ with the corresponding Gibbs measure initial data and prove invariance of the Gibbs measure under the flow, thereby resolving a question posed by Oh, Robert, and Wang (2019). Our physical space approach hinges on developing a novel $L^\infty$-based well-posedness theory for wave equations with exponential-type nonlinearities, going beyond the classical $L^2$-based framework. This refinement allows us to fully leverage structural properties of Gaussian multiplicative chaos. As a by-product of our method, we also obtain an improved well-posedness theory for the hyperbolic Liouville model.