## Abstract

In this note, we study the hyperbolic stochastic damped sine-Gordon equation (SdSG), with a parameter β2 > 0, and its associated Gibbs dynamics on the two-dimensional torus. After introducing a suitable renormalization, we first construct the Gibbs measure in the range 0 < β2 < 4π via the variational approach due to Barashkov-Gubinelli (2018). We then prove almost sure global well-posedness and invariance of the Gibbs measure under the hyperbolic SdSG dynamics in the range 0 < β2 < 2π. Our construction of the Gibbs measure also yields almost sure global well-posedness and invariance of the Gibbs measure for the parabolic sine-Gordon model in the range 0 < β2 < 4π.

Original language | English |
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Pages (from-to) | 1450-1466 |

Number of pages | 17 |

Journal | Proceedings of the Royal Society of Edinburgh Section A: Mathematics |

Volume | 151 |

Issue number | 5 |

Early online date | 16 Sept 2020 |

DOIs | |

Publication status | Published - Oct 2021 |

### Bibliographical note

Publisher Copyright:Copyright © The Author(s), 2020. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh.

## Keywords

- dynamical sine-Gordon model
- Gaussian multiplicative chaos
- Gibbs measure
- renormalization
- Stochastic sine-Gordon equation
- white noise

## ASJC Scopus subject areas

- General Mathematics