Path-by-path well-posedness of nonlinear diffusion equations with multiplicative noise (Q2656191)

In this interesting work, the authors consider a class of stochastic porous media and fast diffusion equations with multiplicative noise, and nonnegative initial data, for which they prove path-by-path well-posedness. The general equation is reformulated in the kinetic setting and the notion of pathwise kinetic solution is introduced; this approach is motivated by the theory of stochastic viscosity solutions for fully-nonlinear second-order SPDEs and stochastic scalar conservation laws. The existence of such solutions is established in the vanishing viscosity limit through and a priori estimates are obtained for an approximate problem with regularized noise. As a result, a corresponding random dynamical system exists when for example a fractional Brownian motion appears in the noise definition.