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

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, R"ockner; JDE, 2011], [Barbu, R"ockner; JDE, 2018+], and [Gess, AoP, 2014].