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Invariant Gibbs dynamics for the hyperbolic sinh-Gordon model

Justin Forlano, Younes Zine

Abstract

We study the hyperbolic defocusing sinh-Gordon model with parameter $β^2>0$ and its associated Gibbs dynamics on the two-dimensional torus. We establish global well-posedness of the model for a certain range of parameters $β^2>0$ 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.

Invariant Gibbs dynamics for the hyperbolic sinh-Gordon model

Abstract

We study the hyperbolic defocusing sinh-Gordon model with parameter and its associated Gibbs dynamics on the two-dimensional torus. We establish global well-posedness of the model for a certain range of parameters 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 -based well-posedness theory for wave equations with exponential-type nonlinearities, going beyond the classical -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.
Paper Structure (18 sections, 34 theorems, 392 equations)

This paper contains 18 sections, 34 theorems, 392 equations.

Table of Contents

  1. Introduction
  2. The sinh-Gordon model
  3. Setup and main result
  4. Overview of the proofs
  5. Preliminaries
  6. Notations
  7. Function spaces, embeddings and linear estimates
  8. Analytic estimates
  9. Basic product estimates
  10. Kernel estimates
  11. Kernel estimates with additional damping
  12. Stochastic objects
  13. The stochastic convolution
  14. The Gaussian multiplicative chaos
  15. The modified Gaussian multiplicative chaos
  16. ...and 3 more sections

Key Result

Theorem 1.1

Let $0 < \beta^2 < \frac{6\pi}{5}$ and $\iota \neq 0$. Then, the stochastic damped sinh-Gordon equation liouville3 is almost surely locally well-posed with respect to the Gaussian free field $\vec{\mu}_1$ defined in series. More precisely, there exists an $\vec{\mu}_1$-almost surely positive time $T

Theorems & Definitions (67)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Remark 1.4
  • Remark 1.5
  • Remark 1.6
  • Remark 1.7
  • Lemma 2.1
  • Definition 2.2
  • Lemma 2.3
  • ...and 57 more