Path-by-path well-posedness of nonlinear diffusion equations with multiplicative noise

Abstract

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]].