an:06537679
Zbl 1374.35385
Marahrens, Daniel; Otto, Felix
On annealed elliptic Green's function estimates
EN
Math. Bohem. 140, No. 4, 489-506 (2015).
00352209
2015
j
35Q55 35A01
stochastic homogenization; elliptic equation; Green's function on \(\mathbb{Z}^d\); annealed estimate
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. \textit{T. Delmotte} and \textit{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 \(\langle |\nabla_x \nabla_y G(x,y)|\rangle\).
Zbl 1083.60082