Homogenization of a semilinear parabolic PDE with locally periodic coefficients : a probabilistic approach
ESAIM: Probability and Statistics, Tome 11 (2007), pp. 385-411

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

DOI : 10.1051/ps:2007026
Classification : 35B27, 60H30, 60J60, 60J35
Keywords: homogenization, nonlinear parabolic PDE, Poisson equation, diffusion approximation, backward SDE 
@article{PS_2007__11__385_0,
     author = {Bench\'erif-Madani, Abdellatif and Pardoux, \'Etienne},
     title = {Homogenization of a semilinear parabolic {PDE} with locally periodic coefficients : a probabilistic approach},
     journal = {ESAIM: Probability and Statistics},
     pages = {385--411},
     year = {2007},
     publisher = {EDP Sciences},
     volume = {11},
     doi = {10.1051/ps:2007026},
     mrnumber = {2339300},
     language = {en},
     url = {https://www.numdam.org/articles/10.1051/ps:2007026/}
}
TY  - JOUR
AU  - Benchérif-Madani, Abdellatif
AU  - Pardoux, Étienne
TI  - Homogenization of a semilinear parabolic PDE with locally periodic coefficients : a probabilistic approach
JO  - ESAIM: Probability and Statistics
PY  - 2007
SP  - 385
EP  - 411
VL  - 11
PB  - EDP Sciences
UR  - https://www.numdam.org/articles/10.1051/ps:2007026/
DO  - 10.1051/ps:2007026
LA  - en
ID  - PS_2007__11__385_0
ER  - 
%0 Journal Article
%A Benchérif-Madani, Abdellatif
%A Pardoux, Étienne
%T Homogenization of a semilinear parabolic PDE with locally periodic coefficients : a probabilistic approach
%J ESAIM: Probability and Statistics
%D 2007
%P 385-411
%V 11
%I EDP Sciences
%U https://www.numdam.org/articles/10.1051/ps:2007026/
%R 10.1051/ps:2007026
%G en
%F PS_2007__11__385_0
Benchérif-Madani, Abdellatif; Pardoux, Étienne. Homogenization of a semilinear parabolic PDE with locally periodic coefficients : a probabilistic approach. ESAIM: Probability and Statistics, Tome 11 (2007), pp. 385-411. doi: 10.1051/ps:2007026

[1] G. Barles and E. Lesigne, SDE, BSDE and PDE. Pitman Res. Notes Math. 364 (1997) 47-80. | Zbl

[2] A. Bencherif-Madani and E. Pardoux, Homogenization of a diffusion with locally periodic coefficients. Sém. Prob. XXXVIII, LNM 1857 (2003) 363-392. | Zbl

[3] A. Bencherif-Madani and E. Pardoux, Locally periodic Homogenization. Asymp. Anal. 39 (2004) 263-279. | Zbl

[4] A. Bensoussan, L. Boccardo and F. Murat, Homogenization of elliptic equations with principal part not in divergence form and Hamiltonian with quadratic growth. Comm. Pure. Appl. Math. 39 (1986) 769-805. | Zbl

[5] R. Buckdahn, Y. Hu and S. Peng, Probabilistic approach to homogenization of viscosity solutions of parabolic PDEs. NoDEA Nonlinear Diff. Eq. Appl. 6 (1999) 395-411. | Zbl

[6] M.G. Crandall, H. Ishii and P.R. Lions, User's guide to viscosity solutions of second order partial differential equations. Bull. A.M.S. 27 (1992) 1-67. | Zbl

[7] F. Delarue, On the existence and uniqueness of solutions to FBSDEs in a non-degenerate case. Stoch. Proc. Appl. 99 (2002) 209-286. | Zbl

[8] F. Delarue, Auxiliary SDEs for homogenization of quasilinear PDEs with periodic coefficients. Ann. Prob. 32 (2004) 2305-2361. | Zbl

[9] A. Jakubowski, A non-Skorohod topology on the Skorohod space. Elec. J. Prob. 2 (1997) 1-21. | Zbl

[10] T. Kurtz, Random time changes and convergence in distribution under the Meyer-Zheng conditions. Ann. Prob. 19 (1991) 1010-1034. | Zbl

[11] P.A. Meyer and W.A. Zheng, Tightness criteria for laws of semimartingales. Anal. I. H. P. 20 (1984) 353-372. | Zbl | Numdam

[12] E. Pardoux, Homogenization of linear and semilinear second order Parabolic PDEs with periodic coefficients: -a probabilistic approach. J. Func. Anal. 167 (1999a) 498-520. | Zbl

[13] 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. | Zbl

Cité par Sources :