Hostname: page-component-cd9895bd7-dk4vv Total loading time: 0 Render date: 2024-12-27T11:44:55.705Z Has data issue: false hasContentIssue false
  • English
  • Français

Homogenization of a semilinear parabolic PDE with locally periodiccoefficients: a probabilistic approach

Published online by Cambridge University Press:  17 August 2007

Abdellatif Benchérif-Madani  and
Étienne Pardoux
Abdellatif Benchérif-Madani
Affiliation:
Université Ferhat Abbas, Fac. Sciences, Dépt. Maths., Sétif 19000, Algeria; lotfi_madani@yahoo.fr
Étienne Pardoux
Affiliation:
CMI, LATP – CNRS and Université de Provence, 39, rue F. Joliot Curie, 13453 Marseille Cedex 13, France; pardoux@latp.unimrs.fr
Get access

Abstract

In this paper, a singular semi-linear parabolic PDE with locally periodiccoefficients is homogenized. We substantially weaken previous assumptions onthe coefficients. In particular, we prove new ergodic theorems. We show thatin such a weak setting on the coefficients, the proper statement of thehomogenization property concerns viscosity solutions, though we need abounded Lipschitz terminal condition.

Keywords

Homogenizationnonlinear parabolic PDEPoisson equationdiffusion approximationbackward SDE.
Type
Research Article
Information
ESAIM: Probability and Statistics , Volume 11: Special Issue: "Stochastic analysis and mathematical finance" in honor of Nicole El Karoui's 60th birthdaySpecial Issue: "Stochastic analysis and mathematical finance" in honor of Nicole El Karoui's 60th birthday , June 2007 , pp. 385 - 411
Copyright
© EDP Sciences, SMAI, 2007

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Barles, G. and Lesigne, E., BSDE, SDE and PDE. Pitman Res. Notes Math. 364 (1997) 47–80.
A. Bencherif-Madani and E. Pardoux, Homogenization of a diffusion with locally periodic coefficients. Sém. Prob. XXXVIII, LNM 1857 (2003) 363–392.
Bencherif-Madani, A. and Pardoux, E., Locally periodic Homogenization. Asymp. Anal. 39 (2004) 263–279.
Bensoussan, A., Boccardo, L. and Murat, F., Homogenization of elliptic equations with principal part not in divergence form and Hamiltonian with quadratic growth. Comm. Pure. Appl. Math. 39 (1986) 769–805. CrossRef
Buckdahn, R., Hu, Y. and Peng, S., Probabilistic approach to homogenization of viscosity solutions of parabolic PDEs. NoDEA Nonlinear Diff. Eq. Appl. 6 (1999) 395–411. CrossRef
Crandall, M.G., Ishii, H. and Lions, P.R., User's guide to viscosity solutions of second order partial differential equations. Bull. A.M.S. 27 (1992) 1–67. CrossRef
Delarue, F., On the existence and uniqueness of solutions to FBSDEs in a non-degenerate case. Stoch. Proc. Appl. 99 (2002) 209–286. CrossRef
Delarue, F., Auxiliary SDEs for homogenization of quasilinear PDEs with periodic coefficients. Ann. Prob. 32 (2004) 2305–2361. CrossRef
Jakubowski, A., A non-Skorohod topology on the Skorohod space. Elec. J. Prob. 2 (1997) 1–21. CrossRef
Kurtz, T., Random time changes and convergence in distribution under the Meyer-Zheng conditions. Ann. Prob. 19 (1991) 1010–1034. CrossRef
Meyer, P.A. and Zheng, W.A., Tightness criteria for laws of semimartingales. Anal. I. H. P. 20 (1984) 353–372.
Pardoux, E., Homogenization of linear and semilinear second order Parabolic PDEs with periodic coefficients: -a probabilistic approach. J. Func. Anal. 167 (1999a) 498–520. CrossRef
E. Pardoux, BSDEs, weak convergence and homogenization of semilinear PDEs, in Nonlinear analysis, Differential Equations and Control, F.H. Clarke and R.J. Stern Eds., Kluwer Acad. Pub. (1999b) 503–549.