We consider the two-dimensional stochastic damped nonlinear wave equation (SdNLW) with the cubic nonlinearity, forced by a space-time white noise. In particular, we investigate the limiting behavior of solutions to SdNLW with regularized noises and establish triviality results in the spirit of the work by Hairer, Ryser, and Weber (2012). More precisely, without renormalization of the nonlinearity, we establish the following two limiting behaviors; (i) in the strong noise regime, we show that solutions to SdNLW with regularized noises tend to 0 as the regularization is removed and (ii) in the weak noise regime, we show that solutions to SdNLW with regularized noises converge to a solution to a deterministic damped nonlinear wave equation with an additional mass term.

中文翻译：

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二维随机非线性波动方程的平凡性评注

我们考虑具有三次非线性的二维随机阻尼非线性波动方程 (SdNLW)，由时空白噪声强制。特别是，我们研究了具有正则化噪声的 SdNLW 解决方案的限制行为，并根据 Hairer、Ryser 和 Weber（2012 年）的工作精神建立了微不足道的结果。更准确地说，在没有非线性重整化的情况下，我们建立了以下两个限制行为；(i) 在强噪声状态下，我们表明正则化噪声的 SdNLW 解随着正则化被去除而趋向于 0 和 (ii) 在弱噪声状态下，我们表明具有正则化噪声的 SdNLW 的解收敛到一个解具有附加质量项的确定性阻尼非线性波动方程。