Nous établissons des résultats d’homogénéisation d’équations aux dérivées partielles paraboliques linéaires et semi–linéaires, sous une hypothèse d’hypoellipticité de l’opérateur aux dérivées partielles du second ordre, au lieu de l’hypothèse usuelle d’ellipticité. Notre méthode de démonstration est essentiellement probabiliste.
We establish homogenization results for both linear and semilinear partial differential equations of parabolic type, when the linear second order PDE operator satisfies a hypoellipticity asumption, rather than the usual ellipticity condition. Our method of proof is essentially probabilistic.
@article{AFST_2007_6_16_2_253_0,
author = {Di\'edhiou, Alassane and Pardoux, \'Etienne},
title = {Homogenization of periodic semilinear hypoelliptic {PDEs}},
journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
pages = {253--283},
publisher = {Universit\'e Paul Sabatier, Institut de math\'ematiques},
address = {Toulouse},
volume = {Ser. 6, 16},
number = {2},
year = {2007},
doi = {10.5802/afst.1148},
zbl = {1131.35304},
mrnumber = {2331541},
language = {en},
url = {http://www.numdam.org/articles/10.5802/afst.1148/}
}
TY - JOUR AU - Diédhiou, Alassane AU - Pardoux, Étienne TI - Homogenization of periodic semilinear hypoelliptic PDEs JO - Annales de la Faculté des sciences de Toulouse : Mathématiques PY - 2007 SP - 253 EP - 283 VL - 16 IS - 2 PB - Université Paul Sabatier, Institut de mathématiques PP - Toulouse UR - http://www.numdam.org/articles/10.5802/afst.1148/ DO - 10.5802/afst.1148 LA - en ID - AFST_2007_6_16_2_253_0 ER -
%0 Journal Article %A Diédhiou, Alassane %A Pardoux, Étienne %T Homogenization of periodic semilinear hypoelliptic PDEs %J Annales de la Faculté des sciences de Toulouse : Mathématiques %D 2007 %P 253-283 %V 16 %N 2 %I Université Paul Sabatier, Institut de mathématiques %C Toulouse %U http://www.numdam.org/articles/10.5802/afst.1148/ %R 10.5802/afst.1148 %G en %F AFST_2007_6_16_2_253_0
Diédhiou, Alassane; Pardoux, Étienne. Homogenization of periodic semilinear hypoelliptic PDEs. Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 16 (2007) no. 2, pp. 253-283. doi : 10.5802/afst.1148. http://www.numdam.org/articles/10.5802/afst.1148/
[1] De Arcangelis (R.), Serra Cassano (F.).— On the homogenization of degenerate elliptic equations in divergence form, J. Math. Pures Appl. 71, p. 119-138, (1992). | MR | Zbl
[2] Bellieud (M.), Bouchitté (G.).— Homogénéisation de problèmes elliptiques dégénérés, CRAS Sér. I Math. 327, p. 787-792, (1998). | MR | Zbl
[3] Biroli (M.), Mosco (U.), Tchou (N.).— Homogenization for degenerate operators with periodical coefficients with respect to the Heisenberg group, CRAS Sér. I Math. 322, p. 439-44, (1996). | MR | Zbl
[4] Brézis (H.).— Analyse fonctionnelle, Théorie et applications, Dunod, Paris, (1999). | MR | Zbl
[5] Delarue (F.).— Auxiliary SDEs for homogenization of quasilinear PDEs with periodic coefficients, Ann. Probab. 32, p. 2305-2361, (2004). | MR | Zbl
[6] Engström (J.), Persson (L.-E.), Piatnitski (A.), Wall (P.).— Homogenization of random degenerated nonlinear monotone operators, Preprint. | MR
[7] Friedman (A.).— Foundations of Modern Analysis, Dover Publications, Inc. (1970). | MR | Zbl
[8] Jakubowski (A.).— A non–Skorohod topology on the Skorohod space, Elec. J. of Probability, 2, (1997). | MR | Zbl
[9] Jurdjevic (V.).— Geometric Control Theory, Cambridge Univ. Press, (1977). | MR | Zbl
[10] Meyer (P. A.), Zheng (W. A.).— Tightness criteria for laws of semi–martingales, Ann. Instit. Henri Poincaré 20, p. 353-372, (1984). | Numdam | MR | Zbl
[11] Michel (D.), Pardoux (É.).— An Introduction to Malliavin calculus and some of its applications, in Recent advances in Stochastic Calculus, J.S. Baras, V. Mirelli eds., p. 65-104, Springer (1990). | MR | Zbl
[12] Pardoux (É.), Veretennikov (A. Yu.).— On Poisson equation and diffusion approximation 1, Ann. Probab. 29, p. 1061-1085, (2001). | MR | Zbl
[13] Pardoux (É.).— Homogenization of Linear and Semilinear Second Order Parabolic PDEs with Periodic Coefficients: A Probabilistic Approch, Journal of Functional Analysis 167, p. 498-520, (1999). | MR | Zbl
[14] Pardoux (É.).— BSDEs, weak convergence and homogenization of semilinear PDEs, in “ Nonlinear Analysis, Differential Equations and Control” (F. H. Clarke and R. J. Stern, EDs.), p. 503-549, Kluwer Acad. Pub., (1999). | Zbl
[15] Paronetto (F.).— Homogenization of degenerate elliptic–parabolic equations, Asymptot. Anal. 37, 21–56, 2004. | MR | Zbl
[16] Strook (D.), and Varadhan (S.R.S.).— On the support of diffusion processes with applications to the maximum principle. Proceedings of sith Berkeley Symposium on Mathematical Statistics and Probability, Vol. III, p. 333-360. | Zbl
[17] Treves (F.).— Introduction to Pseudodifferential and Fourier Integral Operators, Vol. 1, The Univ. series in Mathematics, Plenum Press, NY, (1980). | MR | Zbl
Cité par Sources :
