| Abstrakt: |
Abstract: 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. |
