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Alassane Diédhiou; Étienne Pardoux
Homogenization of periodic semilinear hypoelliptic PDEs
Annales de la faculté des sciences de Toulouse Sér. 6, 16 no. 2 (2007), p. 253-283, doi: 10.5802/afst.1148
Article PDF | Reviews MR 2331541 | Zbl 1131.35304
[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 1170248 | Zbl 0678.35036
[2] Bellieud (M.), Bouchitté (G.).— Homogénéisation de problèmes elliptiques dégénérés, CRAS Sér. I Math. 327, p. 787-792, (1998). MR 1659998 | Zbl 0920.35024
[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 1381780 | Zbl 0851.47046
[4] Brézis (H.).— Analyse fonctionnelle, Théorie et applications, Dunod, Paris, (1999). MR 697382 | Zbl 0511.46001
[5] Delarue (F.).— Auxiliary SDEs for homogenization of quasilinear PDEs with periodic coefficients, Ann. Probab. 32, p. 2305-2361, (2004). Article | MR 2078542 | Zbl 1073.35021
[6] Engström (J.), Persson (L.-E.), Piatnitski (A.), Wall (P.).— Homogenization of random degenerated nonlinear monotone operators, Preprint. MR 2242396
[7] Friedman (A.).— Foundations of Modern Analysis, Dover Publications, Inc. (1970). MR 275100 | Zbl 0557.46001
[8] Jakubowski (A.).— A non–Skorohod topology on the Skorohod space, Elec. J. of Probability, 2, (1997). MR 1475862 | Zbl 0890.60003
[9] Jurdjevic (V.).— Geometric Control Theory, Cambridge Univ. Press, (1977). MR 1425878 | Zbl 0940.93005
[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 771895 | Zbl 0551.60046
[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 1255163 | Zbl 0729.60046
[12] Pardoux (É.), Veretennikov (A. Yu.).— On Poisson equation and diffusion approximation 1, Ann. Probab. 29, p. 1061-1085, (2001). Article | MR 1872736 | Zbl 1029.60053
[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 1716206 | Zbl 0935.35010
[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 0959.60049
[15] Paronetto (F.).— Homogenization of degenerate elliptic–parabolic equations, Asymptot. Anal. 37, 21–56, 2004. MR 2035361 | Zbl 1052.35025
[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 0255.60056
[17] Treves (F.).— Introduction to Pseudodifferential and Fourier Integral Operators, Vol. 1, The Univ. series in Mathematics, Plenum Press, NY, (1980). MR 597144 | Zbl 0453.47027
Alassane Diédhiou; Étienne Pardoux
Homogenization of periodic semilinear hypoelliptic PDEs
Annales de la faculté des sciences de Toulouse Sér. 6, 16 no. 2 (2007), p. 253-283, doi: 10.5802/afst.1148
Article PDF | Reviews MR 2331541 | Zbl 1131.35304
Résumé - Abstract
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.
Bibliography
[2] Bellieud (M.), Bouchitté (G.).— Homogénéisation de problèmes elliptiques dégénérés, CRAS Sér. I Math. 327, p. 787-792, (1998). MR 1659998 | Zbl 0920.35024
[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 1381780 | Zbl 0851.47046
[4] Brézis (H.).— Analyse fonctionnelle, Théorie et applications, Dunod, Paris, (1999). MR 697382 | Zbl 0511.46001
[5] Delarue (F.).— Auxiliary SDEs for homogenization of quasilinear PDEs with periodic coefficients, Ann. Probab. 32, p. 2305-2361, (2004). Article | MR 2078542 | Zbl 1073.35021
[6] Engström (J.), Persson (L.-E.), Piatnitski (A.), Wall (P.).— Homogenization of random degenerated nonlinear monotone operators, Preprint. MR 2242396
[7] Friedman (A.).— Foundations of Modern Analysis, Dover Publications, Inc. (1970). MR 275100 | Zbl 0557.46001
[8] Jakubowski (A.).— A non–Skorohod topology on the Skorohod space, Elec. J. of Probability, 2, (1997). MR 1475862 | Zbl 0890.60003
[9] Jurdjevic (V.).— Geometric Control Theory, Cambridge Univ. Press, (1977). MR 1425878 | Zbl 0940.93005
[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 771895 | Zbl 0551.60046
[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 1255163 | Zbl 0729.60046
[12] Pardoux (É.), Veretennikov (A. Yu.).— On Poisson equation and diffusion approximation 1, Ann. Probab. 29, p. 1061-1085, (2001). Article | MR 1872736 | Zbl 1029.60053
[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 1716206 | Zbl 0935.35010
[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 0959.60049
[15] Paronetto (F.).— Homogenization of degenerate elliptic–parabolic equations, Asymptot. Anal. 37, 21–56, 2004. MR 2035361 | Zbl 1052.35025
[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 0255.60056
[17] Treves (F.).— Introduction to Pseudodifferential and Fourier Integral Operators, Vol. 1, The Univ. series in Mathematics, Plenum Press, NY, (1980). MR 597144 | Zbl 0453.47027
