Homogenization of semilinear PDEs with discontinuous averaged coefficients

Khaled Bahlali (Université de Toulon)
A Elouaflin (Université de Cocody)
Etienne Pardoux (Université de Provence)


We study the asymptotic behavior of solutions of semilinear PDEs. Neither periodicity nor ergodicity will be assumed. On the other hand, we assume that the coecients have averages in the Cesaro sense. In such a case, the averaged coecients could be discontinuous. We use a probabilistic approach based on weak convergence of the associated backward stochastic dierential equation (BSDE) in the Jakubowski $S$-topology to derive the averaged PDE. However, since the averaged coecients are discontinuous, the classical viscosity solution is not dened for the averaged PDE. We then use the notion of "$L_p$-viscosity solution" introduced in [7]. The existence of $L_p$-viscosity solution to the averaged PDE is proved here by using BSDEs techniques.

Full Text: Download PDF | View PDF online (requires PDF plugin)

Pages: 477-499

Publication Date: February 22, 2009

DOI: 10.1214/EJP.v14-627


  1. Bahlali, Khaled. Existence and uniqueness of solutions for BSDEs with locally Lipschitz coefficient. Electron. Comm. Probab. 7 (2002), 169--179 (electronic). MR1937902 (2003h:60081)
  2. Benchérif-Madani, Abdellatif; Pardoux, Étienne. Homogenization of a semilinear parabolic PDE with locally periodic coefficients: a probabilistic approach. ESAIM Probab. Stat. 11 (2007), 385--411 (electronic). MR2339300
  3. Bensoussan, Alain; Lions, Jacques-Louis; Papanicolaou, George. Asymptotic analysis for periodic structures.Studies in Mathematics and its Applications, 5. North-Holland Publishing Co., Amsterdam-New York, 1978. xxiv+700 pp. ISBN: 0-444-85172-0 MR0503330 (82h:35001)
  4. Buckdahn, Rainer; Ichihara, Naoyuki. Limit theorem for controlled backward SDEs and homogenization of Hamilton-Jacobi-Bellman equations. Appl. Math. Optim. 51 (2005), no. 1, 1--33. MR2101380 (2005j:60131)
  5. Buckdahn, Rainer; Hu, Ying; Peng, Shige. Probabilistic approach to homogenization of viscosity solutions of parabolic PDEs. NoDEA Nonlinear Differential Equations Appl. 6 (1999), no. 4, 395--411. MR1736544 (2000m:35019)
  6. Buckdahn, Rainer; Hu, Ying. Probabilistic approach to homogenizations of systems of quasilinear parabolic PDEs with periodic structures. Nonlinear Anal. 32 (1998), no. 5, 609--619. MR1612034 (99g:35017)
  7. Caffarelli, L.; Crandall, M. G.; Kocan, M.; Swipolhk ech, A. On viscosity solutions of fully nonlinear equations with measurable ingredients. Comm. Pure Appl. Math. 49 (1996), no. 4, 365--397. MR1376656 (97a:35051)
  8. Crandall, M.G., Kocan, M., Lions, P. L., Swiech, A. Existence results for boundary problems for uniformly elliptic and parabolic fully nonlinear equations. Electronic Journal of Differential equations., No. 1-20, (1999).
  9. Delarue, François. Auxiliary SDEs for homogenization of quasilinear PDEs with periodic coefficients. Ann. Probab. 32 (2004), no. 3B, 2305--2361. MR2078542 (2005e:35015)
  10. Backward stochastic differential equations.Papers from the study group held at the University of Paris VI, Paris, 1995--1996.Edited by Nicole El Karoui and Laurent Mazliak.Pitman Research Notes in Mathematics Series, 364. Longman, Harlow, 1997. ii+221 pp. ISBN: 0-582-30733-3 MR1752671 (2000k:60003)
  11. Essaky, El Hassan; Ouknine, Youssef. Averaging of backward stochastic differential equations and homogenization of partial differential equations with periodic coefficients. Stoch. Anal. Appl. 24 (2006), no. 2, 277--301. MR2204713 (2007b:60142)
  12. Freidlin, Mark. Functional integration and partial differential equations.Annals of Mathematics Studies, 109. Princeton University Press, Princeton, NJ, 1985. x+545 pp. ISBN: 0-691-08354-1; 0-691-08362-2 MR0833742 (87g:60066)
  13. Ichihara, Naoyuki. A stochastic representation for fully nonlinear PDEs and its application to homogenization. J. Math. Sci. Univ. Tokyo 12 (2005), no. 3, 467--492. MR2192225 (2007a:60040)
  14. Jakubowski, Adam. A non-Skorohod topology on the Skorohod space. Electron. J. Probab. 2 (1997), no. 4, 21 pp. (electronic). MR1475862 (98k:60046)
  15. Khasminskii, R.; Krylov, N. On averaging principle for diffusion processes with null-recurrent fast component. Stochastic Process. Appl. 93 (2001), no. 2, 229--240. MR1828773 (2002e:60131)
  16. Jikov, V. V.; Kozlov, S. M.; Oleĭnik, O. A. Homogenization of differential operators and integral functionals.Translated from the Russian by G. A. Yosifian [G. A. Iosifʹyan].Springer-Verlag, Berlin, 1994. xii+570 pp. ISBN: 3-540-54809-2 MR1329546 (96h:35003b)
  17. Krylov, N. V. Controlled diffusion processes.Translated from the Russian by A. B. Aries.Applications of Mathematics, 14. Springer-Verlag, New York-Berlin, 1980. xii+308 pp. ISBN: 0-387-90461-1 MR0601776 (82a:60062)
  18. Krylov, N. V. On weak uniqueness for some diffusions with discontinuous coefficients. Stochastic Process. Appl. 113 (2004), no. 1, 37--64. MR2078536 (2005e:60119)
  19. Lejay, Antoine. A probabilistic approach to the homogenization of divergence-form operators in periodic media. Asymptot. Anal. 28 (2001), no. 2, 151--162. MR1869029 (2002j:60112)
  20. Meyer, P.-A.; Zheng, W. A. Tightness criteria for laws of semimartingales. Ann. Inst. H. Poincaré Probab. Statist. 20 (1984), no. 4, 353--372. MR0771895 (86c:60008)
  21. Pankov, Alexander. $G$-convergence and homogenization of nonlinear partial differential operators.Mathematics and its Applications, 422. Kluwer Academic Publishers, Dordrecht, 1997. xiv+249 pp. ISBN: 0-7923-4720-X MR1482803 (99a:35017)
  22. Pardoux, E. Backward stochastic differential equations and viscosity solutions of systems of semilinear parabolic and elliptic PDEs of second order, in Stochastic analysis and related topics VI, Geilo 1996, Progr. Probab., vol. 42, Birkhauser, Boston, MA, pp. 79--127, (1998).
  23. Pardoux, Étienne. BSDEs, weak convergence and homogenization of semilinear PDEs. Nonlinear analysis, differential equations and control (Montreal, QC, 1998), 503--549, NATO Sci. Ser. C Math. Phys. Sci., 528, Kluwer Acad. Publ., Dordrecht, 1999. MR1695013 (2000e:60096)
  24. Pardoux, Étienne. Homogenization of linear and semilinear second order parabolic PDEs with periodic coefficients: a probabilistic approach. J. Funct. Anal. 167 (1999), no. 2, 498--520. MR1716206 (2000j:60079)
  25. Pardoux, E.; Veretennikov, A. Yu. Averaging of backward stochastic differential equations, with application to semi-linear PDE's. Stochastics Stochastics Rep. 60 (1997), no. 3-4, 255--270. MR1467720 (98d:60125)

Creative Commons License
This work is licensed under a Creative Commons Attribution 3.0 License.