BSDE driven by Dirichlet process and semi-linear parabolic PDE. Application to homogenization

Stochastic Processes and their Applications

Volumn 97, Issue 1, 2002, Pages 1-39

  Lejay, Antoine ^a  

^a UNIVERSITÉ HENRI POINCARÉ NANCY 1   (France)

Author keywords

BSDE; Divergence form operator; Homogenization; Periodic media; Random media

Indexed keywords

EID: 0038095734     PISSN: 03044149     EISSN: None     Source Type: Journal    
DOI: 10.1016/S0304-4149(01)00124-7     Document Type: Article

References (46)

1. Aronson, D.G., 1968. Non-negative solutions of linear parabolic equation. Ann. Scuola Norm. Sup. Pisa 22, 607-693.
2. Bally, V., Pardoux, É., Stoica, L., 1999. BSDE's associated to semilinear PDE's with measurable coefficients. Université du Maine, preprint.
3. Barles, G., Lesigne, E., 1997. SDE, BSDE and PDE. In: Backward Stochastic Differential Equations. Pitman Research Notes in Mathematics Series, Longman, New York, pp. 47-80.
4. Bensoussan, A., Lions, J.L., Papanicolaou, G., 1978. Asymptotic Analysis for Periodic Structures. North-Holland, Amsterdam.
5. Billingsley, P., 1968. Convergence of Probability Measures. Wiley, New York.
6. Buckdahn, R., Hu, Y., Peng, S., 1999. Probabilistic approach to homogenization of viscosity solutions of parabolic PDEs. NoDEA Nonlinear Differential Equations Appl. 6, 395-411.
7. Castell, F., 2001. Homogenization of random semilinear PDEs. Probab. Theory Related Fields, in press.
8. Chen, Z.Q., Zhao, Z., 1995. Diffusion processes and second order elliptic operators with singular coefficients for lower order terms. Math. Ann. 302 (2), 323-357.
9. El Karoui, N., 1997. Backward stochastic differential equations: a general introduction. In: Backward Stochastic Differential Equations (Pitman Research Notes in Mathematics Series, Longman, New York, pp. 7-26.
10. Ethier, S.N., Kurtz, T.G., 1986. Markov Processes, Characterization and Convergence. Wiley, New York.
11. Freidlin, M., 1996. Markov Processes and Differential Equations: Asymptotic Problems. Lectures in Mathematics, ETH Zürich. Birkhaüser, Basel.
12. Friedman, A.V., 1964. Partial Differential Equations of Parabolic Type. Prentice-Hall, Englewood Cliffs, NJ.
13. Fukushima, M., Oshima, Y., Takeda, M., 1994. Dirichlet Forms and Symmetric Markov Process. De Gruyter, Berlin.
14. Gaudron, G., Pardoux, É., 2001. Convergence en loi d'EDSR et application à l'homogénéisation d'EDP semi-linéaires paraboliques. Ann. Inst. H. Poincaré Probab. Statist. 37 (1), 1-40.
15. Hu, Y., 1997. Stability theorems and homogenization of nonlinear PDEs with periodic structures. In: Backward Stochastic Differential Equations. Pitman Research Notes in Mathematics Series, Longman, New York, pp. 193-205.
16. Hu, Y., Peng, S., 1997. A stability theorem of backward stochastic differential equations and its application. C. R. Acad. Sci. Paris Sr. I Math. 324 (9), 1059-1064.
17. Jacod, J., Shiryaev, A.N., 1987. Limit Theorems for Stochastic Processes. Springer, Berlin.
18. Jakubowski, A., 1997. A non-Skorohod topology on the Skorohod space. Electron. J. Probab. 2 (paper no. 4, pp. 1-21).
19. 1 de Skorohod. Probab. Theory Related Fields 81, 111-137.
20. Jikov, V.V., Kozlov, S.M., Oleinik, O.A., 1994. Homogenization of Differential Operators and Integral Functionals. Springer, Berlin.
21. Krylov, N.V., 1980. Controlled Diffusion Processes, Applications of Mathematics. Springer, New York.
22. Kurtz, T.G., Protter, P., 1995. Weak convergence of stochastic integrals and differential equations. In: Talay, D., Tubaro, L. (Eds.), Probabilistic Models for Nonlinear Partial Differential Equations, Montecatini Terme. Lecture Notes in Mathematics No. 1629, Springer, Berlin, pp. 1-41.
23. Ladyženskaja, O.A., Solonnikov, V.A., Ural'ceva, N.N., 1968. Linear and Quasilinear Equations of Parabolic Type. American Mathematical Society, Providence, RI.
24. Lejay, A., 2000a. Méthodes probabilistes pour l'homogé néisation des opérateurs sous forme-divergence: cas linéaires et semi-linéaires. Ph.D. Thesis, Université de Provence, Marseille, France.
25. Lejay, A., 2000b. A probabilistic representation of the solution of some quasi-linear PDE with a divergence-form operator. Application to existence of weak solutions of FBSDE, submitted for publication.
26. Lejay, A., 2001a. A Probabilistic approach of the homogenization of divergence-form operators in periodic media. Asymptotic Anal., in press.
27. Lejay, A., 2001b. Homogenization of divergence-form operators with lower order terms in random media. Probab. Theory Related Fields 120 (2), 255-276.
28. Littman, W., Stampacchia, G., Weinberger, H.F., 1963. Regular points for elliptic equations with discontinuous coefficients. Ann. Scuola Norm. Sup. Pisa 17 (3), 43-77.
29. Lyons, T., Zhang, T., 1996. Convergence of non-symmetric Dirichlet processes. Stochastics Stochastics Rep. 57, 159-167.
30. Ma, J., Protter, P., Yong, J., 1994. Solving forward backward stochastic differential equations explicitly-a four step scheme. Prob. Theory Related Fields 98, 339-359.
31. Ma, Z., Röckner, M., 1991. Introduction to the Theory of (Non-Symmetric) Dirichlet Forms. Universitext. Springer, Berlin.
32. p-estimate for the gradient of solutions of second order elliptic divergence equations. Ann. Scuola Norm. Sup. Pisa 17, 189-206.
33. Meyer, P.A., Zheng, W.A., 1984. Tightness criteria for laws of semimartingales. Ann. Inst. H. Poincaré Probab. Statist. 20 (4), 353-372.
34. Olla, S., 1994. Homogenization of diffusion processes in random fields. Cours de l'École doctorale de l'École Polytechnique. École Polytechnique, France.
35. Osada, H., 1998. An invariance principle for Markov processes and Brownian particles with singular interaction. Ann. Inst. H. Poincaré Probab. Statist. 34 (2), 217-248.
36. Pardoux, É., 1999a. BSDEs, weak convergence and homogenization of semilinear PDEs. In: Clarke, F.H., Stern, R.J. (Eds.), Non Linear Analysis, Differential Equations and Control. Proceedings Séminaire de Mathématiques Supérieures, Montréal 1998. Kluwer, Dordrecht, pp. 503-549.
37. Pardoux, É., 1999b. Homogenization of linear and semilinear second order PDEs with periodic coefficients: a probabilistic approach. J. Funct. Anal. 167 (2), 498-520.
38. Pardoux, É., Peng, S., 1990. Adapted solution of a backward stochastic differential equation. Systems Control Lett. 14, 55-61.
39. Pardoux, É., Veretennikov, A.Y., 1997. Averaging of backward SDEs, with application to semi-linear PDEs. Stochastics Stochastics Rep. 60, 255-270.
40. Protter, P., 1990. Stochastic Integration and Differential Equation, A New Approach, Applications of Mathematics. Springer, Berlin.
41. Rozkosz, A., 1996a. Stochastic representation of diffusions corresponding to divergence form operators. Stochastic Process. Appl. 63, 11-33.
42. Rozkosz, A., 1996b. Weak convergence of diffusions corresponding to divergence form operator. Stochastics Stochastics Rep. 57, 129-157.
43. Rozkosz, A., Słomiński, L., 1991. On existence and stability of weak solutions of multidimensional stochastic differential equations with measurable coefficients. Stochastic Process. Appl. 37, 187-197.
44. Stroock, D.W., 1988. Diffusion semigroups corresponding to uniformly elliptic divergence form operator. In: Séminaire de Probabilités XXII, Lecture Notes in Mathematics No. 1321, Springer, Berlin, pp. 316-347.
45. Tomisaki, M., 1980. Superposition of diffusion processes. J. Mat. Soc. Japan 32 (5), 671-696.
46. Zeidler, E., 1990. Nonlinear Functional Analysis and its Applications: Nonlinear Monotone Operators, Vol. II/B. Springer, Berlin.