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stochastic homogenization; elliptic equation; Green's function on $\mathbb Z^d$; annealed estimate
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's function $G(t,x,y)$ satisfy optimal annealed estimates which are $L^2$ and $L^1$, respectively, in probability, i.e., they obtained bounds on $\smash {\langle |\nabla _x G(t,x,y)|^2\rangle ^{{1}/{2}}}$ and $\langle |\nabla _x \nabla _y G(t,x,y)|\rangle $. In particular, the elliptic Green's function $G(x,y)$ satisfies optimal annealed bounds. In their recent work, the authors extended these elliptic bounds to higher moments, i.e., $L^p$ in probability for all $p<\infty $. In this note, we present a new argument that relies purely on elliptic theory to derive the elliptic estimates for $\langle |\nabla _x G(x,y)|^2\rangle ^{{1}/{2}}$ and $\langle |\nabla _x \nabla _y G(x,y)|\rangle $.
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