Stochastic homogenization of a porous-medium type equation

We consider the homogenization problem for the stochastic porous-medium type equation ${\text{p}}_t{u}^{\epsilon }=\mathrm{\Delta }f(T\left(\frac{x}{\text{ep}}\right)\text{om},u^{\text{ep}})$, with a well-prepared initial datum, where $f(T(y)\text{om},u)$ is a stationary process, increasing in $u$, on a given probability space $(\text{Om},\mathcal{F},\mu )$ endowed with an ergodic dynamical system $\{T(y):y\in {\mathbb{R}}^N\}$. Differently from the previous literature \cite{afs,fs}, here we do not assume $\text{Om}$ compact. We first show that the weak solution $u^{\text{ep}}$ satisfies a kinetic formulation of the equation, then we exploit the theory of “stochastically two-scale convergence in the mean” developed in \cite{bmw} to show convergence of the kinetic solution to the kinetic solution of an homogenized problem of the form ${\text{p}}_t\bar{u}-\mathrm{\Delta }\bar{f}(\bar{u})=0$. The homogenization result for the weak solutions then follows.