From quenched invariance principle to semigroup convergence with applications to exclusion processes

Consider a random walk on Zd in a translation-invariant and ergodic random environment and starting from the origin. In this short note, assuming that a quenched invariance principle for the opportunely-rescaled walks holds, we show how to derive an L1-convergence of the corresponding semigroups. We then apply this result to obtain a quenched pathwise hydrodynamic limit for the simple symmetric exclusion process on Zd, d ≥ 2, with i.i.d. symmetric nearest-neighbors conductances ωxy ∈ [0, ∞) only satisfying Q(ωxy > 0) > pc, where pc is the critical value for bond percolation.