  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].
