Stochastic homogenization of Hamilton-Jacobi and degenerate Bellman equations in unbounded environments

Journal des Mathematiques Pures et Appliquees

Volumn 97, Issue 5, 2012, Pages 460-504

  Armstrong, Scott N ^a   Souganidis, Panagiotis E ^a  

^a UNIVERSITY OF CHICAGO   (United States)

Author keywords

Poissonian potential; Stochastic homogenization; Viscous Hamilton Jacobi equation

Indexed keywords

EID: 84859722346     PISSN: 00217824     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.matpur.2011.09.009     Document Type: Article

References (35)

1. Akcoglu M.A., Krengel U. Ergodic theorems for superadditive processes. J. Reine Angew. Math. 1981, 323:53-67.
2. Becker M.E. Multiparameter groups of measure-preserving transformations: a simple proof of Wiener's ergodic theorem. Ann. Probab. 1981, 9(3):504-509.
3. Caffarelli L.A., Souganidis P.E. Rates of convergence for the homogenization of fully nonlinear uniformly elliptic pde in random media. Invent. Math. 2010, 180(2):301-360.
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. 2005, 58(3):319-361.
5. Carmona R., Lacroix J. Spectral Theory of Random Schrödinger Operators. Probability and its Applications 1990, Birkhäuser Boston Inc., Boston, MA.
6. Crandall M.G., Ishii H., Lions P.-L. User's guide to viscosity solutions of second order partial differential equations. Bull. Amer. Math. Soc. (N.S.) 1992, 27(1):1-67.
7. Dal Maso G., Modica L. Nonlinear stochastic homogenization. Ann. Mat. Pura Appl. (4) 1986, 144:347-389.
8. Dal Maso G., Modica L. Nonlinear stochastic homogenization and ergodic theory. J. Reine Angew. Math. 1986, 368:28-42.
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, New York. second ed.
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, New York. second ed.
11. Davini A., Siconolfi A. Metric techniques for convex stationary ergodic Hamiltonians. Calc. Var. Partial Differential Equations 2011, 40(3-4):391-421.
12. Evans L.C. Periodic homogenisation of certain fully nonlinear partial differential equations. Proc. Roy. Soc. Edinburgh Sect. A 1992, 120(3-4):245-265.
13. Ishii H. On the equivalence of two notions of weak solutions, viscosity solutions and distribution solutions. Funkcial. Ekvac. 1995, 38(1):101-120.
14. Kesten H. Percolation Theory for Mathematicians. Progress in Probability and Statistics 1982, vol. 2. Birkhäuser, Boston, MA.
15. Kosygina E., Rezakhanlou F., Varadhan S.R.S. Stochastic homogenization of Hamilton-Jacobi-Bellman equations. Comm. Pure Appl. Math. 2006, 59(10):1489-1521.
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. 2008, 61(6):816-847.
17. Kozlov S.M. The averaging method and walks in inhomogeneous environments. Uspekhi Mat. Nauk 1985, 40(2(242)):61-120. 238.
18. Laslett G.M. Mixing of cluster point processes. J. Appl. Probab. 1978, 15(4):715-725.
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. 1989, 283(4):583-630.
20. 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. 2003, 56(10):1501-1524.
22. Lions P.-L., Souganidis P.E. Homogenization of "viscous" Hamilton-Jacobi equations in stationary ergodic media. Comm. Partial Differential Equations 2005, 30(1-3):335-375.
23. Lions P.-L., Souganidis P.E. Stochastic homogenization of Hamilton-Jacobi and "viscous"-Hamilton-Jacobi equations with convex nonlinearities-revisited. Commun. Math. Sci. 2010, 8(2):627-637.
24. Papanicolaou G.C., Varadhan S.R.S. Boundary value problems with rapidly oscillating random coefficients. Colloq. Math. Soc. János Bolyai 1981, vol. 27:835-873. North-Holland, Amsterdam.
25. Papanicolaou G.C., Varadhan S.R.S. Diffusions with random coefficients. Statistics and Probability: Essays in Honor of C.R. Rao 1982, 547-552. North-Holland, Amsterdam.
26. F. Rassoul-Agha, T. Seppäläinen, A. Yilmaz, Quenched free energy and large deviations for random walks in random potentials, preprint, . arxiv:1104.3110.
27. Rezakhanlou F., Tarver J.E. Homogenization for stochastic Hamilton-Jacobi equations. Arch. Ration. Mech. Anal. 2000, 151(4):277-309.
28. R.W. Schwab, Stochastic homogenization for some nonlinear integro-differential equations, preprint, . arxiv:1101.6052.
29. Schwab R.W. Stochastic homogenization of Hamilton-Jacobi equations in stationary ergodic spatio-temporal media. Indiana Univ. Math. J. 2009, 58(2):537-581.
30. Souganidis P.E. Stochastic homogenization of Hamilton-Jacobi equations and some applications. Asymptot. Anal. 1999, 20(1):1-11.
31. Sznitman A.-S. Brownian asymptotics in a Poissonian environment. Probab. Theory Related Fields 1993, 95(2):155-174.
32. Sznitman A.-S. Brownian motion with a drift in a Poissonian potential. Comm. Pure Appl. Math. 1994, 47(10):1283-1318.
33. Sznitman A.-S. Shape theorem, Lyapunov exponents, and large deviations for Brownian motion in a Poissonian potential. Comm. Pure Appl. Math. 1994, 47(12):1655-1688.
34. Sznitman A.-S. Brownian Motion, Obstacles and Random Media. Springer Monographs in Mathematics 1998, Springer-Verlag, Berlin.
35. Zhikov V.V., Kozlov S.M., Oleinik O.A. Averaging of parabolic operators. Tr. Mosk. Mat. Obs. 1982, 45:182-236.