Stochastic Homogenization for Some Nonlinear Integro-Differential Equations

Communications in Partial Differential Equations

Volumn 38, Issue 2, 2013, Pages 171-198

  Schwab, Russell W ^a  

^a MICHIGAN STATE UNIVERSITY   (United States)

Author keywords

Homogenization; Jump processes; Levy processes; Nonlocal elliptic equations; Obstacle problem; Optimal stochastic control

Indexed keywords

EID: 84872528029     PISSN: 03605302     EISSN: 15324133     Source Type: Journal    
DOI: 10.1080/03605302.2012.741176     Document Type: Article

References (39)

1. Akcoglu, M.A., Krengel, U. (1981). Ergodic theorems for superadditive processes. J. Reine Angew. Math. 223: 53-67.
2. Barles, G., Chasseigne, E., Imbert, C. (2008). On the dirichlet problem for second-order elliptic integro-differential equations. Indiana Univ. Math. J. 57: 213-246.
3. Barles, G., Imbert, C. (2008). Second-order elliptic integro-differential equations: viscosity solutions' theory revisited. Ann. Inst. H. Poincaré Anal. Non Linéaire 25: 567-585.
4. Bensoussan, A., Blankenship, G. (1988). Controlled diffusions in a random medium. Stochastics 24: 87-120.
5. Bensoussan, A., Lions, J.-L., Papanicolaou, G. (1978). Asymptotic Analysis for Periodic Structures. Studies in Mathematics and its Applications, Vol. 5. Amsterdam: North-Holland Publishing Co.
6. Caffarelli, L., Crandall, M.G., Kocan, M., Swiȩch, A. (1996). On viscosity solutions of fully nonlinear equations with measurable ingredients. Comm. Pure Appl. Math.49:365-397.
7. Caffarelli, L., Mellet, A. (2008). Random homogenization of fractional obstacle problems. Netw. Heterog. Media 3: 523-554.
8. Caffarelli, L., Mellet, A. (2009). Random homogenization of an obstacle problem. Ann. Inst. H. Poincaré Anal. Non Linéaire 26: 375-395.
9. Caffarelli, L.A., Silvestre, L. (2011). Regularity results for nonlocal equations by approximation. Arch. Ration. Mech. Anal. 200: 59-88.
10. Caffarelli, L.A., Silvestre, L. (2009). Regularity theory for fully nonlinear integro-differential equations. Comm. Pure Appl. Math. 62: 597-638.
11. Caffarelli, L.A., Silvestre, L. (2009). Regularity theory for fully nonlinear integro-differential equations. Comm. Pure Appl. Math. 62: 597-638.
12. Caffarelli, L.A., Souganidis, P.E. (2010). Rates of convergence for the homogenization of fully non-linear uniformly elliptic pde in random media. Invent. Math. 180: 301-360.
13. Caffarelli, L.A., Souganidis, P.E., Wang, L. (2005). Homogenization of fully nonlinear, uniformly elliptic and parabolic partial differential equations in stationary ergodic media. Comm. Pure Appl. Math. 58: 319-361.
14. Chan, T. (1999). Pricing contingent claims on stocks driven by Lévy processes. Ann. Appl. Probab. 9: 504-528.
15. Crandall, M.G., Ishii, H., Lions, P.-L. (1992). User's guide to viscosity solutions of second order partial differential equations. Bull. Amer. Math. Soc. 27: 1-67.
16. Dal Maso, G., Modica, L. (1986). Nonlinear stochastic homogenization and ergodic theory. J. Reine Angew. Math. 368: 28-42.
17. Evans, L.C. (1992). Periodic homogenisation of certain fully nonlinear partial differential equations. Proc. Roy. Soc. Edinburgh Sect. A 120: 245-265.
18. Focardi, M. (2010). Aperiodic fractional obstacle problems. Adv. Math. 225: 3502-3544.
19. Gilbarg, D., Trudinger, N.S. (2001). Elliptic Partial Differential Equations of Second Order. Classics in Mathematics. Berlin: Springer-Verlag.
20. Guillen, N., Schwab, R.W. (2010). Aleksandroff-Bakelmann-Pucci type estimates for integro-differential equations. Arch. Rat. Mech. Anal. 206: 111-157.
21. Hanson, F.B. (2007). Applied Stochastic Processes and Control for Jump-Diffusions. Advances in Design and Control, Vol. 13. Philadelphia: Society for Industrial and Applied Mathematics.
22. Jikov, V.V., Kozlov, S.M., Oleǐnik, O.A. (1994). Homogenization of Differential Operators and Integral Functionals. Berlin: Springer-Verlag.
23. Kosygina, E., Rezakhanlou, F., Varadhan, S.R.S. (2006). Stochastic homogenization of Hamilton-Jacobi-Bellman equations. Comm. Pure Appl. Math. 59: 1489-1521.
24. Kosygina, E., Varadhan, S.R.S. (2008). Homogenization of Hamilton-Jacobi-Bellman equations with respect to time-space shifts in a stationary ergodic medium. Comm. Pure Appl. Math. 61: 816-847.
25. Kozlov, S.M. (1979). The averaging of random operators. Mat. Sb. 109: 188-202.
26. Lions, P.L., Papanicolaou, G., Varadhan, S.R.S. Homogenization of Hamilton-Jacobi equations. unpublished.
27. Lions, P.-L., Souganidis, P.E. (2003). Correctors for the homogenization of Hamilton-Jacobi equations in the stationary ergodic setting. Comm. Pure Appl. Math. 56: 1501-1524.
28. Lions, P.-L., Souganidis, P.E. (2005). Homogenization of "viscous" Hamilton-Jacobi equations in stationary ergodic media. Comm. Part. Diff. Eqs.30:335-375.
29. Lions, P.-L., Souganidis, P.E. (2010). Stochastic homogenization of Hamilton-Jacobi and "viscous"-Hamilton-Jacobi equations with convex nonlinearities-revisited. Comm. Math. Sci. 8: 627-637.
30. Nolen, J., Novikov, A. (2011). Homogenization of the g-equation with incompressible random drift in two dimensions. Comm. Math. Sci. 9: 561-582.
31. Øksendal, B., Sulem, A. (2007). Applied Stochastic Control of Jump Diffusions. Berlin: Springer.
32. Øksendal, B., Sulem, A. (2009). Risk indifference pricing in jump diffusion markets. Math. Finance 19: 619-637.
33. Papanicolaou, G.C., Varadhan, S.R.S. (1981). Boundary value problems with rapidly oscillating random coefficients. In: Random Fields, Vols. I, II. Colloq. Math. Soc., Vol. 27. Amsterdam: North-Holland, pp. 835-873.
34. Papanicolaou, G.C., Varadhan, S.R.S. (1982). Diffusions with random coefficients. In: Statistics and Probability: Essays in Honor of C. R. Rao. Amsterdam: North-Holland, pp. 547-552.
35. Pham, H. (1998). Optimal stopping of controlled jump diffusion processes: A viscosity solution approach. J. Math. Systems Estim. Control 8: 27 pp.
36. Rezakhanlou, F., Tarver, J.E. (2000). Homogenization for stochastic Hamilton-Jacobi equations. Arch. Ration. Mech. Anal. 151: 277-309.
37. Schwab, R.W. (2009). Stochastic homogenization of Hamilton-Jacobi equations in stationary ergodic spatio-temporal media. Indiana Univ. Math. J. 58: 537-581.
38. Schwab, R.W. (2010). Periodic homogenization for nonlinear integro-differential equations. SIAM J. Math. Anal. 42: 2652-2680.
39. Souganidis, P.E. (1999). Stochastic homogenization of Hamilton-Jacobi equations and some applications. Asympt. Anal. 20: 1-11.