We investigate mean dynamics and invariant measures for a multi-stochastic discrete sine-Gordon equation driven by random viscosity, stochastic forces, and infinite-dimensional nonlinear noise. We first show the existence of a unique solution when the random viscosity is bounded and the nonlinear diffusion of noise is locally Lipschitz continuous, which leads to the existence of a mean random dynamical system. We then prove that such a mean random dynamical system possesses a unique weak pullback mean random attractor in the Bochner space. Finally, we show the existence of an invariant measure. Some difficulties arise from dealing with the term of random viscosity in all uniform estimates (including the tail-estimate) of solutions, which lead to the tightness of a family of distribution laws of solutions.

中文翻译：

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具有随机粘性和非线性噪声的多随机正弦-戈登格的动力学和不变测度

我们研究了由随机粘度、随机力和无限维非线性噪声驱动的多随机离散正弦-戈登方程的平均动力学和不变测度。我们首先展示了当随机粘度有界且噪声的非线性扩散是局部 Lipschitz 连续时唯一解的存在，这导致平均随机动力系统的存在。然后我们证明这样的平均随机动力系统在 Bochner 空间中具有独特的弱回拉平均随机吸引子。最后，我们证明了不变测度的存在。在解决方案的所有统一估计（包括尾估计）中处理随机粘度项会产生一些困难，这会导致解决方案分布规律族的紧密性。