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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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