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 ${\beta }^2>0$, and its associated Gibbs dynamics on the two-dimensional torus. After introducing a suitable renormalization, we first construct the Gibbs measure in the range $0<{\beta }^2<4\pi$ 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 $0<{\beta }^2<2\pi$. 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 $0<{\beta }^2<4\pi$.

15 pages

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