Previous |  Up |  Next


Full entry | Fulltext not available (moving wall 12 months)      Feedback
stochastic homogenization; elliptic equation; Green's function on $\mathbb Z^d$; annealed estimate
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 $.
[1] Aronson, D. G.: Bounds for the fundamental solution of a parabolic equation. Bull. Am. Math. Soc. 73 (1967), 890-896. DOI 10.1090/S0002-9904-1967-11830-5 | MR 0217444 | Zbl 0153.42002
[2] Delmotte, T.: Parabolic Harnack inequality and estimates of Markov chains on graphs. Rev. Mat. Iberoam. 15 (1999), 181-232. DOI 10.4171/RMI/254 | MR 1681641 | Zbl 0922.60060
[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 10.1007/s00440-005-0430-y | MR 2198017 | Zbl 1083.60082
[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, (2015). MR 3418538 | Zbl 1326.39015
[5] Marahrens, D., Otto, F.: Annealed estimates on the Green function. (to appear) in Probab. Theory Relat. Fields, DOI 10.1007/s00440-014-0598-0 | MR 3418749
[6] Nash, J. F.: Continuity of solutions of parabolic and elliptic equations. Am. J. Math. 80 (1958), 931-954. DOI 10.2307/2372841 | MR 0100158 | Zbl 0096.06902
Partner of
EuDML logo