Stochastic integrability of heat-kernel bounds for random walks in a balanced random environment

We consider random walks in a balanced i.i.d. random environment in $Z^d$ for $d\ge 2$ and the corresponding discrete non-divergence form difference operators. We first obtain an exponential integrability of the heat kernel bounds. We then prove the optimal diffusive decay of the semigroup generated by the heat kernel for $d\ge 3$. As a consequence, we deduce a functional central limit theorem for the environment viewed from the particle.