Periodic approximations of the ergodic constants in the stochastic homogenization of nonlinear second-order (degenerate) equations
Annales de l'I.H.P. Analyse non linéaire, Tome 32 (2015) no. 3, p. 571-591
Le texte intégral des articles récents est réservé aux abonnés de la revue. Consulter l'article sur le site de la revue
We prove that the effective nonlinearities (ergodic constants) obtained in the stochastic homogenization of Hamilton–Jacobi, “viscous” Hamilton–Jacobi and nonlinear uniformly elliptic pde are approximated by the analogous quantities of appropriate “periodizations” of the equations. We also obtain an error estimate, when there is a rate of convergence for the stochastic homogenization.
@article{AIHPC_2015__32_3_571_0,
     author = {Cardaliaguet, Pierre and Souganidis, Panagiotis E.},
     title = {Periodic approximations of the ergodic constants in the stochastic homogenization of nonlinear second-order (degenerate) equations},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     publisher = {Elsevier},
     volume = {32},
     number = {3},
     year = {2015},
     pages = {571-591},
     doi = {10.1016/j.anihpc.2014.01.007},
     zbl = {1320.35040},
     mrnumber = {3353701},
     language = {en},
     url = {http://www.numdam.org/item/AIHPC_2015__32_3_571_0}
}
Cardaliaguet, Pierre; Souganidis, Panagiotis E. Periodic approximations of the ergodic constants in the stochastic homogenization of nonlinear second-order (degenerate) equations. Annales de l'I.H.P. Analyse non linéaire, Tome 32 (2015) no. 3, pp. 571-591. doi : 10.1016/j.anihpc.2014.01.007. http://www.numdam.org/item/AIHPC_2015__32_3_571_0/

[1] S.N. Armstrong, P. Cardaliaguet, P.E. Souganidis, Error estimates and convergence rates for the stochastic homogenization of Hamilton–Jacobi equations, J. Am. Math. Soc. 27 no. 2 (2014), 479 -540 | Zbl 1286.35023

[2] S.N. Armstrong, P. Cardaliaguet, Quantitative stochastic homogenization of viscous Hamilton–Jacobi equations, arXiv:1312.7593 | MR 3285244 | Zbl 1320.35032

[3] S.N. Armstrong, C.K. Smart, Regularity and stochastic homogenization of fully nonlinear equations without uniform ellipticity, arXiv:1208.4570 | MR 3265174 | Zbl 1315.35019

[4] S.N. Armstrong, C.K. Smart, Quantitative stochastic homogenization of elliptic equations in nondivergence form, arXiv:1306.5340 | MR 3269637 | Zbl 1304.35714

[5] S.N. Armstrong, P.E. Souganidis, Stochastic homogenization of Hamilton–Jacobi and degenerate Bellman equations in unbounded environments, J. Math. Pures Appl. 97 (2012), 460 -504 | MR 2914944 | Zbl 1246.35029

[6] S.N. Armstrong, P.E. Souganidis, Stochastic homogenization of level-set convex Hamilton–Jacobi equations, Int. Math. Res. Not. 15 (2013), 3420 -3449 | MR 3089731 | Zbl 1319.35003

[7] G. Barles, Solutions de viscosité des équations de Hamilton–Jacobi, Math. Appl. (Berlin) vol. 17 , Springer-Verlag, Paris (1994) | MR 1613876

[8] A. Bourgeat, A. Piatnitski, Approximations of effective coefficients in stochastic homogenization, Ann. Inst. Henri Poincaré Probab. Stat. 40 no. 2 (2004), 153 -165 | MR 2044813 | Zbl 1058.35023

[9] L.A. Caffarelli, X. Cabre, Fully Nonlinear Elliptic Partial Differential Equations, Amer. Math. Soc. (1997)

[10] L.A. Caffarelli, P.E. Souganidis, Rates of convergence for the homogenization of fully nonlinear uniformly elliptic pde in random media, Invent. Math. 180 no. 2 (2010), 301 -360 | MR 2609244 | Zbl 1192.35048

[11] L.A. Caffarelli, P.E. Souganidis, A rate of convergence for monotone finite difference approximations to fully nonlinear, uniformly elliptic PDEs, Commun. Pure Appl. Math. 61 no. 1 (2008), 1 -17 | MR 2361302 | Zbl 1140.65075

[12] L.A. Caffarelli, P.E. Souganidis, L. Wang, Homogenization of fully nonlinear, uniformly elliptic and parabolic partial differential equations in stationary ergodic media, Commun. Pure Appl. Math. 58 no. 3 (2005), 319 -361 | MR 2116617 | Zbl 1063.35025

[13] I. Capuzzo Dolcetta, H. Ishii, On the rate of convergence in homogenization of Hamilton–Jacobi equations, Indiana Univ. Math. J. 50 no. 3 (2001), 1113 -1129 | MR 1871349 | Zbl 1256.35003

[14] P. Cardaliaguet, P.E. Souganidis, Homogenization and enhancement for the G-equation in random environment, Commun. Pure Appl. Math. 66 no. 10 (2013), 1582 -1628 | MR 3084699 | Zbl 1284.60126

[15] M.G. Crandall, H. Ishii, P.-L. Lions, User's guide to viscosity solutions of second order partial differential equations, Bull. Am. Math. Soc. (N.S.) 27 no. 1 (1992), 1 -67 | Zbl 0755.35015

[16] G. Dal Maso, L. Modica, Nonlinear stochastic homogenization, Ann. Mat. Pura Appl. 4 (1986), 347 -389 | MR 870884 | Zbl 0607.49010

[17] G. Dal Maso, L. Modica, Nonlinear stochastic homogenization and ergodic theory, J. Reine Angew. Math. 368 (1986), 28 -42 | MR 850613 | Zbl 0582.60034

[18] X. Guo, O. Zeitouni, Quenched invariance principle for random walks in balanced random environment, arXiv:1003.3494 | MR 2875757 | Zbl 1239.60092

[19] S.M. Kozlov, The averaging method and walks in inhomogeneous environments, Usp. Mat. Nauk 40 no. 2 (1985), 61 -120 | MR 786087 | Zbl 0592.60054

[20] E. Kosygina, F. Rezakhanlou, S.R.S. Varadhan, Stochastic homogenization of Hamilton–Jacobi–Bellman equations, Commun. Pure Appl. Math. 59 no. 10 (2006), 1489 -1521 | MR 2248897 | Zbl 1111.60055

[21] E. Kosygina, S.R.S. Varadhan, Homogenization of Hamilton–Jacobi–Bellman equations with respect to time–space shifts in a stationary ergodic medium, Commun. Pure Appl. Math. 61 no. 6 (2008), 816 -847 | MR 2400607 | Zbl 1144.35008

[22] G. Lawler, Weak convergence of random walk in random environments, Commun. Math. Phys. 87 (1982), 81 -87 | MR 680649 | Zbl 0502.60056

[23] J. Lin, On the stochastic homogenization of fully nonlinear uniformly parabolic equations in stationary ergodic spatio-temporal media, arXiv:1307.4743 | MR 3295277 | Zbl 1326.35023

[24] P.-L. Lions, P.E. Souganidis, Correctors for the homogenization of Hamilton–Jacobi equations in the stationary ergodic setting, Commun. Pure Appl. Math. 56 no. 10 (2003), 501 -1524 | MR 1988897 | Zbl 1050.35012

[25] P.-L. Lions, P.E. Souganidis, Homogenization of “viscous” Hamilton–Jacobi equations in stationary ergodic media, Commun. Partial Differ. Equ. 30 no. 1–3 (2005), 335 -375 | MR 2131058 | Zbl 1065.35047

[26] P.-L. Lions, P.E. Souganidis, Stochastic homogenization of Hamilton–Jacobi and “viscous”-Hamilton–Jacobi equations with convex nonlinearities—revisited, Commun. Math. Sci. 8 no. 2 (2010), 627 -637 | MR 2664465 | Zbl 1197.35031

[27] I. Matic, J. Nolen, A sublinear variance bound for solutions of a random Hamilton–Jacobi equation, J. Stat. Phys. 149 no. 2 (2012), 342 -361 | MR 2988962 | Zbl 1273.82061

[28] J. Nolen, A. Novikov, Homogenization of the G-equation with incompressible random drift, Commun. Math. Sci. 9 no. 2 (2011), 561 -582 | MR 2815685 | Zbl 1241.35021

[29] H. Owhadi, Approximation of effective conductivity of ergodic media by periodization, Probab. Theory Relat. Fields 125 (2003), 225 -258 | MR 1961343 | Zbl 1040.60025

[30] G. Papanicolaou, S.R.S. Varadhan, Boundary value problems with rapidly oscillating random coefficients, Random Fields, vols. I, II, Esztergom, 1979, Colloq. Math. Soc. János Bolyai vol. 27 , North-Holland, Amsterdam (1981), 835 -873 | MR 712714 | Zbl 0499.60059

[31] G. Papanicolaou, S.R.S. Varadhan, Diffusions with random coefficients, Statistics and Probability: Essays in Honor of C.R. Rao, North-Holland, Amsterdam (1982), 547 -552 | MR 659505 | Zbl 0486.60076

[32] F. Rezakhanlou, J.E. Tarver, Homogenization for stochastic Hamilton–Jacobi equations, Arch. Ration. Mech. Anal. 151 no. 4 (2000), 277 -309 | MR 1756906 | Zbl 0954.35022

[33] P.E. Souganidis, Stochastic homogenization of Hamilton–Jacobi equations and some applications, Asymptot. Anal. 20 (1999), 1 -11 | MR 1697831 | Zbl 0935.35008

[34] R. Schwab, Stochastic homogenization of Hamilton–Jacobi equations in stationary ergodic spatio-temporal media, Indiana Univ. Math. J. 58 no. 2 (2009), 537 -581 | MR 2514380 | Zbl 1180.35082

[35] V.V. Zhikov, S.M. Kozlov, O. Oleĭnik, Averaging of parabolic operators, Tr. Mosk. Mat. Obŝ. 45 (1982), 182 -236 | MR 704631 | Zbl 0531.35041