Berry–Esseen theorem and quantitative homogenization for the random conductance model with degenerate conductances
Tóm tắt
We study the random conductance model on the lattice
$${\mathbb {Z}}^d$$
, i.e. we consider a linear, finite-difference, divergence-form operator with random coefficients and the associated random walk under random conductances. We allow the conductances to be unbounded and degenerate elliptic, but they need to satisfy a strong moment condition and a quantified ergodicity assumption in form of a spectral gap estimate. As a main result we obtain in dimension
$$d\ge 3$$
quantitative central limit theorems for the random walk in form of a Berry–Esseen estimate with speed
$$t^{-\frac{1}{5}+\varepsilon }$$
for
$$d\ge 4$$
and
$$t^{-\frac{1}{10}+\varepsilon }$$
for
$$d=3$$
. Additionally, in the uniformly elliptic case in low dimensions
$$d=2,3$$
we improve the rate in a quantitative Berry–Esseen theorem recently obtained by Mourrat. As a central analytic ingredient, for
$$d\ge 3$$
we establish near-optimal decay estimates on the semigroup associated with the environment process. These estimates also play a central role in quantitative stochastic homogenization and extend some recent results by Gloria, Otto and the second author to the degenerate elliptic case.
Tài liệu tham khảo
Andres, S., Barlow, M.T., Deuschel, J.-D., Hambly, B.M.: Invariance principle for the random conductance model. Probab. Theory Relat. Fields 156(3–4), 535–580 (2013)
Andres, S., Deuschel, J.-D., Slowik, M.: Invariance principle for the random conductance model in a degenerate ergodic environment. Ann. Probab. 43(4), 1866–1891 (2015)
Andres, S., Deuschel, J.-D., Slowik, M.: Harnack inequalities on weighted graphs and some applications to the random conductance model. Probab. Theory Relat. Fields 164(3–4), 931–977 (2016)
Andres, S., Deuschel, J.-D., Slowik, M.: Heat kernel estimates for random walks with degenerate weights. Electron. J. Probab. 21, 33, 21 (2016)
Andres, S., Deuschel, J.-D., Slowik, M.: Heat kernel estimates and intrinsic metric for random walks with general speed measure under degenerate conductances. Preprint, available at arXiv:1711.11119 (2017)
Armstrong, S., Dario, P.: Elliptic regularity and quantitative homogenization on percolation clusters. ArXiv e-prints, Sept (2016)
Armstrong, S., Kuusi, T., Mourrat, J.-C.: Quantitative stochastic homogenization and large-scale regularity. Preprint, available at arXiv:1705.05300 (2017)
Armstrong, S., Kuusi, T., Mourrat, J.-C.: The additive structure of elliptic homogenization. Invent. Math. 208(3), 999–1154 (2017)
Armstrong, S.N., Mourrat, J.-C.: Lipschitz regularity for elliptic equations with random coefficients. Arch. Ration. Mech. Anal. 219, 255–348 (2016)
Armstrong, S .N., Smart, C .K.: Quantitative stochastic homogenization of convex integral functionals. Ann. Sci. Éc. Norm. Supér. (4) 49(2), 423–481 (2016)
Barlow, M., Burdzy, K., Timár, Á.: Comparison of quenched and annealed invariance principles for random conductance model. Probab. Theory Relat. Fields 164(3–4), 741–770 (2016)
Bella, P., Fehrman, B., Fischer, J., Otto, F.: Stochastic homogenization of linear elliptic equations: higher-order error estimates in weak norms via second-order correctors. SIAM J. Math. Anal. 49(6), 4658–4703 (2017)
Bella, P., Fehrman, B., Otto, F.: A Liouville theorem for elliptic systems with degenerate ergodic coefficients. Ann. Appl. Probab. 28(3), 1379–1422 (2018)
Bella, P., Giunti, A., Otto, F.: Quantitative stochastic homogenization: local control of homogenization error through corrector. In: Mathematics and materials, vol. 23 of IAS/Park City Math. Ser., pp. 301–327. Am. Math. Soc. Providence, RI (2017)
Ben-Artzi, J., Marahrens, D., Neukamm, S.: Moment bounds on the corrector of stochastic homogenization of non-symmetric elliptic finite difference equations. Commun. Partial Differ. Equ. 42(2), 179–234 (2017)
Biskup, M.: Recent progress on the random conductance model. Probab. Surv. 8, 294–373 (2011)
Brascamp, H.J., Lieb, E.H.: On extensions of the Brunn–Minkowski and Prékopa–Leindler theorems, including inequalities for log concave functions, and with an application to the diffusion equation. J. Funct. Anal. 22(4), 366–389 (1976)
de Buyer, P., Mourrat, J.-C.: Diffusive decay of the environment viewed by the particle. Electron. Commun. Probab. 20, 23, 12 (2015)
De Masi, A., Ferrari, P.A., Goldstein, S., Wick, W.D.: An invariance principle for reversible Markov processes. Applications to random motions in random environments. J. Stat. Phys. 55(3–4), 787–855 (1989)
Dembo, A., Funaki, T.: Stochastic interface models. In: Picard, J. (ed.) Lectures on Probability Theory and Statistics. Lecture Notes in Mathematics, vol. 1869. Springer, Berlin, Heidelberg (2005)
Derriennic, Y., Lin, M.: Fractional Poisson equations and ergodic theorems for fractional coboundaries. Israel J. Math. 123, 93–130 (2001)
Fischer, J., Otto, F.: Sublinear growth of the corrector in stochastic homogenization: optimal stochastic estimates for slowly decaying correlations. Stoch. Partial Differ. Equ. Anal. Comput. 5(2), 220–255 (2017)
Flegel, F., Heida, M., Slowik, M.: Homogenization theory for the random conductance model with degenerate ergodic weights and unbounded-range jumps. ArXiv e-prints, Feb (2017)
Giunti, A., Mourrat, J. C.: Quantitative homogenization of degenerate random environments. Ann. de l’Institut Henri Poincaré, Probab. et Statistiques, 54(1), 22–50 (2018)
Gloria, A., Neukamm, S., Otto, F.: A regularity theory for random elliptic operators. arXiv:1409.2678 (2014)
Gloria, A., Neukamm, S., Otto, F.: Quantitative homogenization for correlated coefficient fields (in preparation)
Gloria, A., Neukamm, S., Otto, F.: Quantification of ergodicity in stochastic homogenization: optimal bounds via spectral gap on Glauber dynamics—long version. MPI Leipzig, preprint 3 (2013)
Gloria, A., Neukamm, S., Otto, F.: Quantification of ergodicity in stochastic homogenization: optimal bounds via spectral gap on Glauber dynamics. Invent. Math. 199(2), 455–515 (2015)
Gloria, A., Otto, F.: An optimal variance estimate in stochastic homogenization of discrete elliptic equations. Ann. Probab. 39(3), 779–856 (2011)
Gloria, A., Otto, F.: An optimal error estimate in stochastic homogenization of discrete elliptic equations. Ann. Appl. Probab. 22(1), 1–28 (2012)
Gloria, A., Otto, F.: The corrector in stochastic homogenization: optimal rates, stochastic integrability, and fluctuations. ArXiv e-prints, Oct. (2015)
Gloria, A., Otto, F.: Quantitative results on the corrector equation in stochastic homogenization. J. Eur. Math. Soc. (JEMS) 19(11), 3489–3548 (2017)
Haeusler, E.: On the rate of convergence in the central limit theorem for martingales with discrete and continuous time. Ann. Probab. 16(1), 275–299 (1988)
Helland, I.S.: Central limit theorems for martingales with discrete or continuous time. Scand. J. Stat. 9(2), 79–94 (1982)
Heyde, C.C., Brown, B.M.: On the departure from normality of a certain class of martingales. Ann. Math. Stat. 41, 2161–2165 (1970)
Kipnis, C., Varadhan, S.R.S.: Central limit theorem for additive functionals of reversible Markov processes and applications to simple exclusions. Commun. Math. Phys. 104(1), 1–19 (1986)
Kozlov, S.M.: The averaging of random operators. Mat. Sb. (N.S.), 109(151)(2):188–202, 327 (1979)
Krengel, U.: Ergodic theorems, volume 6 of de Gruyter Studies in Mathematics. Walter de Gruyter & Co., Berlin, (1985). With a supplement by Antoine Brunel
Kumagai, T.: Random walks on disordered media and their scaling limits, volume 2101 of Lecture Notes in Mathematics. Springer, Cham, 2014. Lecture notes from the 40th Probability Summer School held in Saint-Flour, École d’Été de Probabilités de Saint-Flour [Saint-Flour Probability Summer School] (2010)
Lamacz, A., Neukamm, S., Otto, F.: Moment bounds for the corrector in stochastic homogenization of a percolation model. Electron. J. Probab. 20, 30 (2015)
Marahrens, D., Otto, F.: Annealed estimates on the Green function. Probab. Theory Relat. Fields 163(3–4), 527–573 (2015)
Mourrat, J.-C.: Variance decay for functionals of the environment viewed by the particle. In: Annales de l’institut Henri Poincaré (B), vol. 47, pp. 294–327 (2011)
Mourrat, J.-C.: A quantitative central limit theorem for the random walk among random conductances. Electron. J. Probab., 17, 97, 17 (2012)
Mourrat, J.-C.: On the rate of convergence in the martingale central limit theorem. Bernoulli 19(2), 633–645 (2013)
Mourrat, J.-C., Otto, F.: Anchored Nash inequalities and heat kernel bounds for static and dynamic degenerate environments. J. Funct. Anal. 270(1), 201–228 (2016)
Naddaf, A., Spencer, T.: On homogenization and scaling limit of some gradient perturbations of a massless free field. Commun. Math. Phys. 183(1), 55–84 (1997)
Neukamm, S.: An introduction to the qualitative and quantitative theory of homogenization. Interdiscipl. Inform. Sci. 24(1), 1–48 (2018)
Neukamm, S., Schäffner, M., Schlömerkemper, A.: Stochastic homogenization of nonconvex discrete energies with degenerate growth. SIAM J. Math. Anal. 49(3), 1761–1809 (2017)
Papanicolaou, G.C., Varadhan, S.R.S.: Boundary value problems with rapidly oscillating random coefficients. In: Random fields, vol. I, II (Esztergom, 1979), volume 27 of Colloq. Math. Soc. János Bolyai, pp. 835–873. North-Holland, Amsterdam-New York (1981)
Sidoravicius, V., Sznitman, A.S.: Quenched invariance principles for walks on clusters of percolation or among random conductances. Probab. Theory Relat. Fields 129(2), 219–244 (2004)
Yurinskiĭ, V.V.: Averaging of symmetric diffusion in a random medium. Sibirsk. Mat. Zh. 27(4), 167–180, 215 (1986)