×
Armstrong, Scott N.; Souganidis, Panagiotis E.

Stochastic homogenization of Hamilton-Jacobi and degenerate Bellman equations in unbounded environments. (English. French summary) Zbl 1246.35029

J. Math. Pures Appl. (9) 97, No. 5, 460-504 (2012).
The authors deal with the (stochastic) homogenization of Hamilton-Jacobi equations and Bellman equations posed in stationary, ergodic, unbounded media. They obtain in the limit deterministic Hamilton-Jacobi equations and study the properties of the effective Hamiltonian. It seems that the class of problems studied here is motivated by the work of A.-S. Sznitman [Brownian motion, obstacles and random media. Berlin: Springer (1993; Zbl 0973.60003)] on quenched large deviations of the Brownian motion interacting with a Poisson-like potential.
Reviewer: Adrian Muntean (Eindhoven)

Cited in 45 Documents

MSC:

35B27 Homogenization in context of PDEs; PDEs in media with periodic structure
60K37 Processes in random environments
35F21 Hamilton-Jacobi equations

Keywords:

viscous Hamilton-Jacobi equations, Bellman equations, Poisson potential; effective Hamiltonian

Citations:

Zbl 0973.60003
Cite Review PDF
Full Text: DOI arXiv

References:

[1] Akcoglu, M. A.; Krengel, U., Ergodic theorems for superadditive processes, J. Reine Angew. Math., 323, 53-67 (1981) · Zbl 0453.60039
[2] Becker, M. E., Multiparameter groups of measure-preserving transformations: a simple proof of Wienerʼs ergodic theorem, Ann. Probab., 9, 3, 504-509 (1981) · Zbl 0468.28020
[3] Caffarelli, L. A.; Souganidis, P. E., Rates of convergence for the homogenization of fully nonlinear uniformly elliptic pde in random media, Invent. Math., 180, 2, 301-360 (2010) · Zbl 1192.35048
[4] Caffarelli, L. A.; Souganidis, P. E.; Wang, L., Homogenization of fully nonlinear, uniformly elliptic and parabolic partial differential equations in stationary ergodic media, Comm. Pure Appl. Math., 58, 3, 319-361 (2005) · Zbl 1063.35025
[5] Carmona, R.; Lacroix, J., Spectral Theory of Random Schrödinger Operators, Probability and its Applications (1990), Birkhäuser Boston Inc.: Birkhäuser Boston Inc. Boston, MA · Zbl 0717.60074
[6] Crandall, M. G.; Ishii, H.; Lions, P.-L., Userʼs guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc. (N.S.), 27, 1, 1-67 (1992) · Zbl 0755.35015
[7] Dal Maso, G.; Modica, L., Nonlinear stochastic homogenization, Ann. Mat. Pura Appl. (4), 144, 347-389 (1986) · Zbl 0607.49010
[8] Dal Maso, G.; Modica, L., Nonlinear stochastic homogenization and ergodic theory, J. Reine Angew. Math., 368, 28-42 (1986) · Zbl 0582.60034
[9] Daley, D. J.; Vere-Jones, D., An Introduction to the Theory of Point Processes, vol. I, Probability and its Applications (New York) (2003), Springer-Verlag: Springer-Verlag New York · Zbl 1026.60061
[10] Daley, D. J.; Vere-Jones, D., An Introduction to the Theory of Point Processes, vol. II, Probability and its Applications (New York) (2008), Springer: Springer New York · Zbl 1026.60061
[11] Davini, A.; Siconolfi, A., Metric techniques for convex stationary ergodic Hamiltonians, Calc. Var. Partial Differential Equations, 40, 3-4, 391-421 (2011) · Zbl 1213.49036
[12] Evans, L. C., Periodic homogenisation of certain fully nonlinear partial differential equations, Proc. Roy. Soc. Edinburgh Sect. A, 120, 3-4, 245-265 (1992) · Zbl 0796.35011
[13] Ishii, H., On the equivalence of two notions of weak solutions, viscosity solutions and distribution solutions, Funkcial. Ekvac., 38, 1, 101-120 (1995) · Zbl 0833.35053
[14] Kesten, H., Percolation Theory for Mathematicians, Progress in Probability and Statistics, vol. 2 (1982), Birkhäuser: Birkhäuser Boston, MA · Zbl 0522.60097
[15] Kosygina, E.; Rezakhanlou, F.; Varadhan, S. R.S., Stochastic homogenization of Hamilton-Jacobi-Bellman equations, Comm. Pure Appl. Math., 59, 10, 1489-1521 (2006) · Zbl 1111.60055
[16] Kosygina, E.; Varadhan, S. R.S., Homogenization of Hamilton-Jacobi-Bellman equations with respect to time-space shifts in a stationary ergodic medium, Comm. Pure Appl. Math., 61, 6, 816-847 (2008) · Zbl 1144.35008
[17] Kozlov, S. M., The averaging method and walks in inhomogeneous environments, Uspekhi Mat. Nauk, 40, 2(242), 61-120 (1985), 238 · Zbl 0592.60054
[18] Laslett, G. M., Mixing of cluster point processes, J. Appl. Probab., 15, 4, 715-725 (1978) · Zbl 0393.60050
[19] Lasry, J.-M.; Lions, P.-L., Nonlinear elliptic equations with singular boundary conditions and stochastic control with state constraints. I, The model problem, Math. Ann., 283, 4, 583-630 (1989) · Zbl 0688.49026
[20] P.-L. Lions, G.C. Papanicolaou, S.R.S. Varadhan, Homogenization of Hamilton-Jacobi equations, unpublished, preprint, 1987.; P.-L. Lions, G.C. Papanicolaou, S.R.S. Varadhan, Homogenization of Hamilton-Jacobi equations, unpublished, preprint, 1987.
[21] Lions, P.-L.; Souganidis, P. E., Correctors for the homogenization of Hamilton-Jacobi equations in the stationary ergodic setting, Comm. Pure Appl. Math., 56, 10, 1501-1524 (2003) · Zbl 1050.35012
[22] Lions, P.-L.; Souganidis, P. E., Homogenization of “viscous” Hamilton-Jacobi equations in stationary ergodic media, Comm. Partial Differential Equations, 30, 1-3, 335-375 (2005) · Zbl 1065.35047
[23] Lions, P.-L.; Souganidis, P. E., Stochastic homogenization of Hamilton-Jacobi and “viscous”-Hamilton-Jacobi equations with convex nonlinearities—revisited, Commun. Math. Sci., 8, 2, 627-637 (2010) · Zbl 1197.35031
[24] Papanicolaou, G. C.; Varadhan, S. R.S., Boundary value problems with rapidly oscillating random coefficients, (Random Fields, vols. I, II. Random Fields, vols. I, II, Esztergom, 1979. Random Fields, vols. I, II. Random Fields, vols. I, II, Esztergom, 1979, Colloq. Math. Soc. János Bolyai, vol. 27 (1981), North-Holland: North-Holland Amsterdam), 835-873 · Zbl 0499.60059
[25] Papanicolaou, G. C.; Varadhan, S. R.S., Diffusions with random coefficients, (Statistics and Probability: Essays in Honor of C.R. Rao (1982), North-Holland: North-Holland Amsterdam), 547-552 · Zbl 0499.60059
[26] F. Rassoul-Agha, T. Seppäläinen, A. Yilmaz, Quenched free energy and large deviations for random walks in random potentials, preprint, arXiv:1104.3110 [math.PR]; F. Rassoul-Agha, T. Seppäläinen, A. Yilmaz, Quenched free energy and large deviations for random walks in random potentials, preprint, arXiv:1104.3110 [math.PR]
[27] Rezakhanlou, F.; Tarver, J. E., Homogenization for stochastic Hamilton-Jacobi equations, Arch. Ration. Mech. Anal., 151, 4, 277-309 (2000) · Zbl 0954.35022
[28] R.W. Schwab, Stochastic homogenization for some nonlinear integro-differential equations, preprint, arXiv:1101.6052 [math.AP]; R.W. Schwab, Stochastic homogenization for some nonlinear integro-differential equations, preprint, arXiv:1101.6052 [math.AP]
[29] Schwab, R. W., Stochastic homogenization of Hamilton-Jacobi equations in stationary ergodic spatio-temporal media, Indiana Univ. Math. J., 58, 2, 537-581 (2009) · Zbl 1180.35082
[30] Souganidis, P. E., Stochastic homogenization of Hamilton-Jacobi equations and some applications, Asymptot. Anal., 20, 1, 1-11 (1999) · Zbl 0935.35008
[31] Sznitman, A.-S., Brownian asymptotics in a Poissonian environment, Probab. Theory Related Fields, 95, 2, 155-174 (1993) · Zbl 0792.60100
[32] Sznitman, A.-S., Brownian motion with a drift in a Poissonian potential, Comm. Pure Appl. Math., 47, 10, 1283-1318 (1994) · Zbl 0814.60021
[33] Sznitman, A.-S., Shape theorem, Lyapunov exponents, and large deviations for Brownian motion in a Poissonian potential, Comm. Pure Appl. Math., 47, 12, 1655-1688 (1994) · Zbl 0814.60022
[34] Sznitman, A.-S., Brownian Motion, Obstacles and Random Media, Springer Monographs in Mathematics (1998), Springer-Verlag: Springer-Verlag Berlin · Zbl 0973.60003
[35] Zhikov, V. V.; Kozlov, S. M.; Oleĭnik, O. A., Averaging of parabolic operators, Tr. Mosk. Mat. Obs., 45, 182-236 (1982) · Zbl 0531.35041
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.