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On annealed elliptic Green’s function estimates. (English) Zbl 1374.35385
Summary: 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. T. Delmotte and J.-D. Deuschel [Probab. Theory Relat. Fields 133, No. 3, 358–390 (2005; Zbl 1083.60082)] 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 $$\langle |\nabla_x \nabla_y G(x,y)|\rangle$$.

##### MSC:
 35Q55 NLS equations (nonlinear Schrödinger equations) 35A01 Existence problems for PDEs: global existence, local existence, non-existence
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