Stochastic homogenization of nonconvex unbounded integral functionals with generalized Orlicz growth

We consider the homogenization of random integral functionals which are possibly unbounded, that is, the domain of the integrand is not the whole space and may depend on the space-variable. In the vectorial case, we develop a complete stochastic homogenization theory for nonconvex unbounded functionals with convex growth of generalized Orlicz-type, under a standard set of assumptions in the field, in particular a coercivity condition of order $p^{-}>1$, and an upper bound of order $p^{+}<\mathrm{\infty }$. The limit energy is defined in a possibly anisotropic Musielak-Orlicz space, for which approximation results with smooth functions are provided. The proof is based on the localization method of $\mathrm{\Gamma }$-convergence and a careful use of truncation arguments.