| Title:
|
On annealed elliptic Green's function estimates (English) |
| Author:
|
Marahrens, Daniel |
| Author:
|
Otto, Felix |
| Language:
|
English |
| Journal:
|
Mathematica Bohemica |
| ISSN:
|
0862-7959 (print) |
| ISSN:
|
2464-7136 (online) |
| Volume:
|
140 |
| Issue:
|
4 |
| Year:
|
2015 |
| Pages:
|
489-506 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| 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. 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 $. (English) |
| Keyword:
|
stochastic homogenization |
| Keyword:
|
elliptic equation |
| Keyword:
|
Green's function on $\mathbb Z^d$ |
| Keyword:
|
annealed estimate |
| MSC:
|
35A01 |
| MSC:
|
35Q55 |
| idZBL:
|
Zbl 06537679 |
| idMR:
|
MR3432548 |
| DOI:
|
10.21136/MB.2015.144465 |
| . |
| Date available:
|
2015-11-17T20:55:01Z |
| Last updated:
|
2020-07-29 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/144465 |
| . |
| Reference:
|
[1] Aronson, D. G.: Bounds for the fundamental solution of a parabolic equation.Bull. Am. Math. Soc. 73 (1967), 890-896. Zbl 0153.42002, MR 0217444, 10.1090/S0002-9904-1967-11830-5 |
| Reference:
|
[2] Delmotte, T.: Parabolic Harnack inequality and estimates of Markov chains on graphs.Rev. Mat. Iberoam. 15 (1999), 181-232. Zbl 0922.60060, MR 1681641, 10.4171/RMI/254 |
| Reference:
|
[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. Zbl 1083.60082, MR 2198017, 10.1007/s00440-005-0430-y |
| Reference:
|
[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). Zbl 1326.39015, MR 3418538 |
| Reference:
|
[5] Marahrens, D., Otto, F.: Annealed estimates on the Green function.(to appear) in Probab. Theory Relat. Fields, http://dx.doi.org/10.1007/s00440-014-0598-0. MR 3418749, 10.1007/s00440-014-0598-0 |
| Reference:
|
[6] Nash, J. F.: Continuity of solutions of parabolic and elliptic equations.Am. J. Math. 80 (1958), 931-954. Zbl 0096.06902, MR 0100158, 10.2307/2372841 |
| . |
