Invariant Gibbs dynamics for the dynamical sine-Gordon model

Tadahiro Oh; Tristan Robert; Philippe Sosoe; Yuzhao Wang

doi:10.1017/prm.2020.68

Invariant Gibbs dynamics for the dynamical sine-Gordon model

Tadahiro Oh, Tristan Robert, Philippe Sosoe, Yuzhao Wang

Mathematics

Research output: Contribution to journal › Article › peer-review

3 Citations (Scopus)

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
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	https://doi.org/10.1017/prm.2020.68
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

Access to Document

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Cite this

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title = "Invariant Gibbs dynamics for the dynamical sine-Gordon model",

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. ",

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AU - Oh, Tadahiro

AU - Robert, Tristan

AU - Sosoe, Philippe

AU - Wang, Yuzhao

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N2 - 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.

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KW - dynamical sine-Gordon model

KW - Gaussian multiplicative chaos

KW - Gibbs measure

KW - renormalization

KW - Stochastic sine-Gordon equation

KW - white noise

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Invariant Gibbs dynamics for the dynamical sine-Gordon model

Abstract

Bibliographical note

Keywords

ASJC Scopus subject areas

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