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

Consider a random walk on Z(d) 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 L-1-convergence of the corresponding semigroups. We then apply this result to obtain a quenched pathwise hydrodynamic limit for the simple symmetric exclusion process on Z(d), d >= 2, with i.i.d. symmetric nearest-neighbors conductances omega(xy )is an element of [0,infinity) only satisfying Q(omega(xy)>0) > p(c), where p(c) is the critical value for bond percolation.