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On Homogenization Of Elliptic Equations With Random Coefficients

   
@article{EJP65,
	author = {Joseph Conlon and Ali Naddaf},
	title = {On Homogenization Of Elliptic Equations With Random Coefficients},
	journal = {Electron. J. Probab.},
	fjournal = {Electronic Journal of Probability},
	volume = {5},
	year = {2000},
	keywords = {Homogenization, elliptic equations, random environment, Euler-Lagrange equation.},
	abstract = {In this paper, we investigate the rate of convergence of the solution $u_\varepsilon$ of the random elliptic partial difference equation $(\nabla^{\varepsilon *} a(x/\varepsilon,\omega)\nabla^\varepsilon+1)u_\varepsilon(x,\omega)=f(x)$ to the corresponding homogenized solution. Here $x\in\varepsilon Z^d$,  and $\omega\in\Omega$ represents the randomness. Assuming that $a(x)$'s are independent and uniformly elliptic, we shall obtain an upper bound $\varepsilon^\alpha$ for the rate of convergence, where $\alpha$ is a constant which depends on the dimension $d\ge 2$ and the deviation of $a(x,\omega)$ from the identity matrix. We will also show that the (statistical) average   of $u_\varepsilon(x,\omega)$ and its derivatives decay  exponentially  for large $x$.},
	pages = {no. 9, 1-58},
	issn = {1083-6489},
	doi = {10.1214/EJP.v5-65},    
        url = {http://ejp.ejpecp.org/article/view/65}}