The stochastic p(ω,t,x)-Laplace equation with cylindrical Wiener process

We propose the analysis of a non-linear parabolic problem of p(ω,t,x)p(ω,t,x)-Laplace type in the framework of Orlicz Lebesgue and Sobolev spaces with variable random exponents and a stochastic forcing by a cylindrical Wiener process. We give a result of well-posedness: existence, uniqueness and stability of the solution, for additive and multiplicative problems.