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.

中文翻译：

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动态正弦-戈登模型的不变吉布斯动力学

在本文中，我们研究了双曲随机阻尼正弦-戈登方程 (SdSG)，其参数为β2> 0，及其在二维环面上的相关吉布斯动力学。在引入合适的重整化后，我们首先在 0 < 范围内构造 Gibbs 测度β2< 4π通过 Barashkov-Gubinelli (2018) 的变分方法。然后，我们几乎可以肯定地证明了 Gibbs 测度在 0 <β2< 2π. 我们对 Gibbs 测度的构造也为抛物线 sine-Gordon 模型在 0 <β2< 4π.