MiS Preprint Repository

We consider a random, uniformly elliptic coefficient field $a$ on the lattice ${\mathbb{Z}}^d$. The distribution $⟨\cdot ⟩$ 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 $⟨|{\mathrm{\nabla }}_{x}G(t,x,y){|}^2{⟩}^{\frac{1}{2}}$ and $⟨|{\mathrm{\nabla }}_x{\mathrm{\nabla }}_{y}G(t,x,y)|⟩$, see T. Delmotte and J.-D. Deuschel: On estimating the derivatives of symmetric diffusions in stationary random environments, with applications to the $\mathrm{\nabla }\varphi$ 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 $p<\mathrm{\infty }$, 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 [DD_ell] below) for $⟨|{\mathrm{\nabla }}_{x}G(x,y){|}^2{⟩}^{\frac{1}{2}}$ and $⟨|{\mathrm{\nabla }}_x{\mathrm{\nabla }}_{y}G(x,y)|⟩$.