Geodesic
On annealed elliptic Green's function estimates
Mathematica Bohemica, Tome 140 (2015) no. 4, pp. 489-506

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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 $.
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 $.
DOI : 10.21136/MB.2015.144465
Classification : 35A01, 35Q55
Keywords: stochastic homogenization; elliptic equation; Green's function on $\mathbb Z^d$; annealed estimate 
Marahrens, Daniel; Otto, Felix. On annealed elliptic Green's function estimates. Mathematica Bohemica, Tome 140 (2015) no. 4, pp. 489-506. doi: 10.21136/MB.2015.144465
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[1] Aronson, D. G.: Bounds for the fundamental solution of a parabolic equation. Bull. Am. Math. Soc. 73 (1967), 890-896. | DOI | MR | Zbl

[2] Delmotte, T.: Parabolic Harnack inequality and estimates of Markov chains on graphs. Rev. Mat. Iberoam. 15 (1999), 181-232. | DOI | MR | Zbl

[3] Delmotte, T., Deuschel, J.-D.: On estimating the derivatives of symmetric diffusions in stationary random environment, with applications to $\nabla\varphi$ interface model. Probab. Theory Relat. Fields 133 (2005), 358-390. | DOI | MR | Zbl

[4] Lamacz, A., Neukamm, S., Otto, F.: Moment bounds for the corrector in stochastic homogenization of a percolation model. Electron J. Probab. 20 Article 106, 30 pages, http://ejp.ejpecp.org/article/view/3618 (2015). | MR | Zbl

[5] Marahrens, D., Otto, F.: Annealed estimates on the Green function. (to appear) in Probab. Theory Relat. Fields, | DOI | MR

[6] Nash, J. F.: Continuity of solutions of parabolic and elliptic equations. Am. J. Math. 80 (1958), 931-954. | DOI | MR | Zbl

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