On annealed elliptic Green function estimates

We consider a random, uniformly elliptic coefficient field $a$ on the lattice $\mathbb{Z}^d$. The distribution $\langle \cdot \rangle$ of the coefficient field is assumed to be stationary. Delmotte and Deuschel showed that the gradient and second mixed derivative of the parabolic Green function $G(t,x,y)$ satisfy optimal annealed estimates which are $L^2$ resp. $L^1$ in probability, i.e. they obtained bounds on $\langle |\nabla_x G(t,x,y)|^2 \rangle^{\frac{1}{2}}$ and $\langle |\nabla_x \nabla_y G(t,x,y)| \rangle$, see T. Delmotte and J.-D. Deuschel: On estimating the derivatives of symmetric diffusions in stationary random environments, with applications to the $\nabla\phi$ interface model, Probab. Theory Relat. Fields 133 (2005), 358--390. In particular, the elliptic Green function $G(x,y)$ satisfies optimal annealed bounds. In a recent work, the authors extended these elliptic bounds to higher moments, i.e. $L^p$ in probability for all $p<\infty$, see D. Marahrens and F. Otto: {Annealed estimates on the Green function}, arXiv:1304.4408 (2013). In this note, we present a new argument that relies purely on elliptic theory to derive the elliptic estimates (see Proposition 1.2 below) for $\langle |\nabla_x G(x,y)|^2 \rangle^{\frac{1}{2}}$ and $\langle |\nabla_x \nabla_y G(x,y)| \rangle$.