Homogenization of "viscous" Hamilton-Jacobi equations in stationary ergodic media

Communications in Partial Differential Equations

Volumn 30, Issue 1-3, 2005, Pages 335-375

  Lions, Pierre Louis ^a,b   Souganidis, Panagiotis E ^b  

^a l'alimentation et l'Environnement   (France)

^b UNIVERSITY OF TEXAS AT AUSTIN   (United States)

Author keywords

Homogenization; Random media; Stationary ergodic media; Stochastic homogenization; Viscous Hamilton Jacobi equations

Indexed keywords

EID: 17444412675     PISSN: 03605302     EISSN: 15324133     Source Type: Journal    
DOI: 10.1081/PDE-200050077     Document Type: Article

References (42)

1. Barles, G. (1991). A weak Bernstein method for fully non-linear eliptic equations. Dif. Int. Equations 4:241-262.
2. Bensoussan, A., Blakenship, G. (1988). Controlled diffusions in a random medium. Stochastics 24:87-120.
3. Barles, G., Souganidis, P. E. (1994). A remark on the asymptotic behavior of the solution of the KPP equation. C.R. Acad. Sci. Paris, Série I 319:679-684.
4. Bourgeat, A., Piatnitski, A. (2004). Approximations of effective coefficients in stochastic homogenization. Ann. Inst. H. Poincaré, Prob. et Stat. 40:153-165.
5. Caffarelli, L. A. (1999). A note on nonlinear homogenization. Comm. Pure Appl. Math. 52:829-838.
6. Castell, F. (2001). Homogenization of random semilinear PDEs. Prob. Theory Relat. Fields 121:492-524.
7. Carmona, R., Lacrois, J. (1990). Spectral Theory of Random Schrödinger Operators. Probability and Its Applications, Boston: Birkhäuser.
8. Crandall, M. G., Ishii, H., Lions, P.-L. (1992). User's guide to viscosity solutions of second order partial differential equations. Bull. AMS 27:1-67.
9. Caffarelli, L. A., Lions, P.-L., Souganidis, P. E. In preparation.
10. Caffarelli, L. A., Souganidis, P. E., Wang, L. (2005). Stochastic homogenization for fully nonlinear, second-order partial differential equations. Comm. Pure Appl. Math. 58:319-361.
11. Dal Maso, G., Modica, L. (1986). Nonlinear stochastic homogenization and ergodic theory. J. Reine Angew. Math. 368:28-42.
12. Evans, L. C. (1989). The perturbed test function method for viscosity solutions of nonlinear PDE. Proc. Roy. Soc. Edinb. A 111:359-375.
13. Evans, L. C. (1992). Periodic homogenization of certain fully nonlinear partial differential equations. Proc. Roy. Soc. Edinb. A 120:245-265.
14. Evans, L. C., Ishii, H. (1985). A PDE approach to some asymptotic problems concerning random differential equations with small noise intensities. Ann. Inst. H. Poincaré: Anal. Non Linéaire 2:1-20.
15. Evans, L. C., Souganidis, P. E. (1989). A PDE approach to geometric optics for certain semilinear parabolic equations. Indiana Univ. Math. J. 38:141-172.
16. Fleming, W. H., Shen, S.-J. (1985). Stochastic variation formula for fundamental solutions of parabolic pde. Applied Math. Optim. 13:193-204.
17. Fleming, W. H., Souganidis, P. E. (1986). PDE-viscosity solution approach to some problems of large deviations. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 13:171-192.
18. Freidlin, M. I. (1968). On factorization of a non-negative definite matrix. In Russian: Prob. Theory Appl. 13:375-378.
19. Freidlin, M. I. (1985). Functional Integration and Partial Differential Equations. Ann. Math. Studies 109. Princeton, NJ: Priceton Univ. Press.
20. Freidlin, M. I., Wentzell, A. D. (1984). Random Perturbations of Dynamical Systems. New York: Springer-Verlag.
21. Ishii, H. (1999). Homogenization of the Cauchy problem for Hamilton-Jacobi Equations. In: Stoch. Analysis, Control, Optimization and Applications. Systems and Control Foundations and Applications. Boston: Birkhäuser, pp. 305-324.
22. Jikov, V. V., Kozlov, S. M., Oleinik, O. A. (1991). Homogenization of Differential Operators and Integral Functions. New York: Springer-Verlag.
23. Kozlov, S. M. (1980). Averaging of random operators. Math. USSR Sbornik. 37:167-180.
24. Karatzas, I., Shreve, S. E. (1990). Brownian Motion and Stochastic Calculus. Graduate Texts in Mathematics 113. Berlin: Springer-Verlag.
25. Kosynga, E., Rezankhanlou, F., Varadhan, S. R. S. Personal communication.
26. Lions, P.-L., (1980). Resolution de problemes elliptiques quasilineaires. Arch Rat Mech. Anal 74:336-353.
27. Lions, P.-L., (1983). Optimal control of diffusion processes and Hamilton-Jacobi-Bellman equations, Part II: Viscosity solutions and uniqueness. Comm. PDE 8:1229-1276.
28. Lions, P.-L., Papanicolaou, G., Varadhan, S. R. S. Homogenization of Hamilton-Jacobi Equations. Preprint.
29. Lions, P.-L., Souganidis, P. E. (2003). Correctors for the homogenization of Hamilton-Jacobi equations in a stationary ergodic setting. Comm. Pure Appl. Math. 56:1501-1524.
30. Lions, P.-L., Souganidis, P. E. Homogenization of degenerate second-order PDE in periodic and almost periodic environments and applications. AHIP, Analyse Nonlineaire, to appear.
31. Lions, P.-L., Souganidis, P. E. In preparation.
32. Majda, A., Souganidis, P. E. (1994). Large scale front dynamics for turbulent reaction-diffusion equations with separated velocity scales. Nonlinearity 7:1-30.
33. Oleinik, O. (1966). Alcuni visultati sulle equazioni lineari e quasilineari ellitico-paraboliche a derivate parziali del second ordine. Rend. Classe Sci. Fis. Mat. Nat. Acad. Naz. Lincei, Sci. 8:40:774-784.
34. Papanicolaou, G., Varadhan, S. R. S. (1979). Boundary value problems with rapidly oscillating random coefficients. Proceed. Colloq. on Random Fields. Rigorous results in statistical mechanics and quantum field theory. J. Fritz, J. L. Lebaritz, D. Szasz, eds. Colloquia Mathematica Societ. Janos Bolyai 10:835-873.
35. Papanicolaou, G., Varadhan, S. R. S. (1981). Diffusion with random coefficients. In: Essays in Statistics and Probability (P. R. Krishnaiah, ed.). North Holland.
36. Rezankhanlou, F., Tarver, J. (2000). Homogenization for stochastic Hamilton-Jacobi equations. Arch. Rat. Mech. Anal. 151:277-309.
37. Sznitzman, A.-S. (1994). Shape theorem, Lyapounov Exponents, and large deviations for Brownian motion in a Poissonian potential. Comm. PAM. 47:1656-1688.
38. Souganidis, P. E. (1999). Stochastic homogenization of Hamilton-Jacobi equations and some applications. Asympt. Anal. 20:1-11.
39. Souganidis, P. E. (1997). Front Propagation: Theory and Applications, CIME Course on Viscosity Solutions and their Applications. Springer-Verlag Lecture Notes Math. 1660. Springer-Verlag.
40. Souganidis, P. E. (2002). Recent developments in the theory of front propagation and its applications. In Modern Methods in Scientific Computing and Applications (A. Bourlioux, M. Gander, eds.). NATO Science Series II, Mathematics, Physics and Chemistry 75, Boston: Kluwer Acad. Publishers.
41. Souganidis, P. E. (1985). Approximation schemes for viscosity solutions of Hamilton-Jacobi equations. J. Diff. Eqs. 56:345-390.
42. Varadhan, S. R. (1984). Large Deviations and Applications, CBMS-NSF Regional Conf. 46.