## Annals of Applied Probability

### Quantitative version of the Kipnis–Varadhan theorem and Monte Carlo approximation of homogenized coefficients

#### Abstract

This article is devoted to the analysis of a Monte Carlo method to approximate effective coefficients in stochastic homogenization of discrete elliptic equations. We consider the case of independent and identically distributed coefficients, and adopt the point of view of the random walk in a random environment. Given some final time $t>0$, a natural approximation of the homogenized coefficients is given by the empirical average of the final squared positions re-scaled by $t$ of $n$ independent random walks in $n$ independent environments. Relying on a quantitative version of the Kipnis–Varadhan theorem combined with estimates of spectral exponents obtained by an original combination of PDE arguments and spectral theory, we first give a sharp estimate of the error between the homogenized coefficients and the expectation of the re-scaled final position of the random walk in terms of $t$. We then complete the error analysis by quantifying the fluctuations of the empirical average in terms of $n$ and $t$, and prove a large-deviation estimate, as well as a central limit theorem. Our estimates are optimal, up to a logarithmic correction in dimension $2$.

#### Article information

Source
Ann. Appl. Probab., Volume 23, Number 4 (2013), 1544-1583.

Dates
First available in Project Euclid: 21 June 2013

https://projecteuclid.org/euclid.aoap/1371834038

Digital Object Identifier
doi:10.1214/12-AAP880

Mathematical Reviews number (MathSciNet)
MR3098442

Zentralblatt MATH identifier
1276.35026

#### Citation

Gloria, Antoine; Mourrat, Jean-Christophe. Quantitative version of the Kipnis–Varadhan theorem and Monte Carlo approximation of homogenized coefficients. Ann. Appl. Probab. 23 (2013), no. 4, 1544--1583. doi:10.1214/12-AAP880. https://projecteuclid.org/euclid.aoap/1371834038

#### References

• [1] Anshelevich, V. V., Khanin, K. M. and Sinaĭ, Y. G. (1982). Symmetric random walks in random environments. Comm. Math. Phys. 85 449–470.
• [2] Barlow, M. T. and Deuschel, J. D. (2010). Invariance principle for the random conductance model with unbounded conductances. Ann. Probab. 38 234–276.
• [3] Berger, N. and Biskup, M. (2007). Quenched invariance principle for simple random walk on percolation clusters. Probab. Theory Related Fields 137 83–120.
• [4] Biskup, M. and Prescott, T. M. (2007). Functional CLT for random walk among bounded random conductances. Electron. J. Probab. 12 1323–1348.
• [5] De Masi, A., Ferrari, P. A., Goldstein, S. and Wick, W. D. (1989). An invariance principle for reversible Markov processes. Applications to random motions in random environments. J. Stat. Phys. 55 787–855.
• [6] Durrett, R. (1996). Probability: Theory and Examples, 2nd ed. Duxbury Press, Belmont, CA.
• [7] Egloffe, A. C., Gloria, A., Mourrat, J. C. and Nguyen, T. N. Unpublished manuscript.
• [8] Gloria, A. (2012). Numerical approximation of effective coefficients in stochastic homogenization of discrete elliptic equations. ESAIM Math. Model. Numer. Anal. 46 1–38.
• [9] Gloria, A. and Mourrat, J. C. (2012). Spectral measure and approximation of homogenized coefficients. Probab. Theory Related Fields 154 287–326.
• [10] Gloria, A. and Otto, F. (2011). An optimal variance estimate in stochastic homogenization of discrete elliptic equations. Ann. Probab. 39 779–856.
• [11] Gloria, A. and Otto, F. (2012). An optimal error estimate in stochastic homogenization of discrete elliptic equations. Ann. Appl. Probab. 22 1–28.
• [12] Hebisch, W. and Saloff-Coste, L. (1993). Gaussian estimates for Markov chains and random walks on groups. Ann. Probab. 21 673–709.
• [13] Kipnis, C. and Varadhan, S. R. S. (1986). Central limit theorem for additive functionals of reversible Markov processes and applications to simple exclusions. Comm. Math. Phys. 104 1–19.
• [14] Klenke, A. (2006). Wahrscheinlichkeitstheorie. Springer, Berlin. (English version appeared as: Probability theory. A comprehensive course, Universitext, Springer, London, 2008.)
• [15] Kozlov, S. M. (1985). The averaging method and walks in inhomogeneous environments. Uspekhi Mat. Nauk 40 61–120, 238.
• [16] Kozlov, S. M. (1987). Averaging of difference schemes. Math. USSR Sbornik 57 351–369.
• [17] Künnemann, R. (1983). The diffusion limit for reversible jump processes on $\mathbf{Z}^{d}$ with ergodic random bond conductivities. Comm. Math. Phys. 90 27–68.
• [18] Mathieu, P. (2008). Quenched invariance principles for random walks with random conductances. J. Stat. Phys. 130 1025–1046.
• [19] Mathieu, P. and Piatnitski, A. (2007). Quenched invariance principles for random walks on percolation clusters. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 463 2287–2307.
• [20] Mourrat, J.-C. (2011). Variance decay for functionals of the environment viewed by the particle. Ann. Inst. Henri Poincaré Probab. Stat. 47 294–327.
• [21] Papanicolaou, G. C. (1983). Diffusions and random walks in random media. In The Mathematics and Physics of Disordered Media (Minneapolis, Minn., 1983). Lecture Notes in Math. 1035 391–399. Springer, Berlin.
• [22] Papanicolaou, G. C. and Varadhan, S. R. S. (1981). Boundary value problems with rapidly oscillating random coefficients. In Random Fields, Vol. I, II (Esztergom, 1979). Colloquia Mathematica Societatis János Bolyai 27 835–873. North-Holland, Amsterdam.
• [23] Sidoravicius, V. and Sznitman, A.-S. (2004). Quenched invariance principles for walks on clusters of percolation or among random conductances. Probab. Theory Related Fields 129 219–244.
• [24] Woess, W. (2000). Random Walks on Infinite Graphs and Groups. Cambridge Tracts in Mathematics 138. Cambridge Univ. Press, Cambridge.