We prove the path-by-path well-posedness of stochastic porous media and fast diffusion equations driven by linear, multiplicative noise. As a consequence, we obtain the existence of a random dynamical system. This solves an open problem raised in [Barbu and Röckner (2011) [4]], [Barbu and Röckner (2018) [6]], and [Gess (2014) [11]].

中文翻译：

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具乘法噪声的非线性扩散方程的逐路径适定性

我们证明了由线性，乘性噪声驱动的随机多孔介质的逐路径适定性和快速扩散方程。结果，我们获得了随机动力学系统的存在。这解决了[Barbu andRöckner（2011）[4]]，[Barbu andRöckner（2018）[6]]和[Gess（2014）[11]]中提出的开放性问题。