We consider the Korteweg--de Vries equation with white noise initial data, posed on the whole real line, and prove the almost sure existence of solutions. Moreover, we show that the solutions obey the group property and follow a white noise law at all times, past or future. As an offshoot of our methods, we also obtain a new proof of the existence of solutions and the invariance of white noise measure in the torus setting.

中文翻译：

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线上 KdV 白噪声的不变性

我们考虑在整条实线上提出的具有白噪声初始数据的 Korteweg--de Vries 方程，并证明解几乎肯定存在。此外，我们证明了解决方案遵循群属性并在任何时候、过去或未来都遵循白噪声定律。作为我们方法的一个分支，我们还获得了解决方案存在的新证明和环面设置中白噪声测度的不变性。