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Invariant Gibbs dynamics for the dynamical sine-Gordon model

Part of: Hyperbolic equations and systems Stochastic analysis

Published online by Cambridge University Press:  16 September 2020

Tadahiro Oh ,
Tristan Robert ,
Philippe Sosoe  and
Yuzhao Wang
Tadahiro Oh
Affiliation:
School of Mathematics, The University of Edinburgh, and The Maxwell Institute for the Mathematical Sciences, James Clerk Maxwell Building, The King's Buildings, Peter Guthrie Tait Road, EdinburghEH9 3FD, UK (hiro.oh@ed.ac.uk)
Tristan Robert
Affiliation:
Fakultät für Mathematik, Universität Bielefeld, Postfach 10 01 31, 33501Bielefeld, Germany University of Rennes, CNRS, IRMAR - UMR 6625, F-35000, Rennes, France (tristan.robert@ens-rennes.fr)
Philippe Sosoe
Affiliation:
Department of Mathematics, Cornell University, 584 Malott Hall, Ithaca, New York14853, USA (psosoe@math.cornell.edu)
Yuzhao Wang
Affiliation:
School of Mathematics, University of Birmingham, Watson Building, Edgbaston BirminghamB15 2TT, United Kingdom (y.wang.14@bham.ac.uk)
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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.

Keywords

Stochastic sine-Gordon equationdynamical sine-Gordon modelrenormalizationwhite noiseGibbs measureGaussian multiplicative chaos

MSC classification

Primary: 35L71: Semilinear second-order hyperbolic equations
Secondary: 60H15: Stochastic partial differential equations
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 151 , Issue 5 , October 2021 , pp. 1450 - 1466
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Copyright
Copyright © The Author(s), 2020. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

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