Invariant Gibbs dynamics for the dynamical sine-Gordon model

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 $0<beta^2<4pi$ 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 $0<beta^2<2pi$. 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 $0<beta^2<4pi$.