Homogenization of Periodic Semilinear Parabolic Degenerate PDEs
Annales de l'I.H.P. Analyse non linéaire, Volume 26 (2009) no. 3, p. 979-998
@article{AIHPC_2009__26_3_979_0,
     author = {Sow, A. B. and Rhodes, R. and Pardoux, \'e.},
     title = {Homogenization of Periodic Semilinear Parabolic Degenerate PDEs},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     publisher = {Elsevier},
     volume = {26},
     number = {3},
     year = {2009},
     pages = {979-998},
     doi = {10.1016/j.anihpc.2008.09.001},
     zbl = {1178.35052},
     mrnumber = {2526412},
     language = {en},
     url = {http://www.numdam.org/item/AIHPC_2009__26_3_979_0}
}
Sow, A. B.; Rhodes, R.; Pardoux, é. Homogenization of Periodic Semilinear Parabolic Degenerate PDEs. Annales de l'I.H.P. Analyse non linéaire, Volume 26 (2009) no. 3, pp. 979-998. doi : 10.1016/j.anihpc.2008.09.001. http://www.numdam.org/item/AIHPC_2009__26_3_979_0/

[1] Bensoussan A., Lions J. L., Papanicolaou G., Asymptotic Analysis for Periodic Structures, North-Holland, 1978. | MR 503330 | Zbl 0404.35001

[2] Briand P., Hu Y., Stability of BSDEs With Random Terminal Time and Homogenization of Semilinear Elliptic PDEs, J. Funct. Anal. 155 (1998) 455-494. | MR 1624569 | Zbl 0912.60081

[3] Buckdahn R., Hu Y., Peng S., Probabilistic Approach to Homogenization of Viscosity Solutions of Parabolic PDEs, NoDEA Nonlinear Differential Equations Appl. 6 (4) (1999) 395-411. | MR 1736544 | Zbl 0953.35017

[4] Crandall M., Ishii H., Lions P. L., User's Guide to Viscosity Solutions of Second Order Partial Differential Equations, Bull. Amer. Math. Soc. 27 (1992) 1-67. | MR 1118699 | Zbl 0755.35015

[5] De Arcangelis R., Serra Cassano F., On the Homogenization of Degenerate Elliptic Equations in Divergence Form, J. Math. Pures Appl. 71 (1992) 119-138. | MR 1170248 | Zbl 0678.35036

[6] Delarue F., On the Existence and Uniqueness of Solutions to FBSDEs in a Non-Degenerate Case, Stochastic Process. Appl. 99 (2) (2002) 209-286. | MR 1901154 | Zbl 1058.60042

[7] Delarue F., Auxiliary SDEs for Homogenization of Quasilinear PDEs With Periodic Coefficients, Ann. Probab. B 32 (3) (2004) 2305-2361. | MR 2078542 | Zbl 1073.35021

[8] Delarue F., Rhodes R., Stochastic Homogenization of Quasilinear PDEs With a Spatial Degeneracy, Asymptot. Anal., in press, available on, http://hal.archives-ouvertes.fr/. | MR 2499193 | Zbl 1180.35591

[9] Diop M. A., Iftimie B., Pardoux E., Piatnitski A., Singular Homogenization With Stationary in Time and Periodic in Space Coefficients, J. Funct. Anal. 231 (2006) 1-46. | MR 2190162 | Zbl 1113.35015

[10] Engström J., Persson L.-E., Piatnitski A., Wall P., Homogenization of Random Degenerated Nonlinear Monotone Operators, Glas. Mat. Ser. III 41 (61) (2006) 101-114. | MR 2242396 | Zbl 1118.35003

[11] Ethier S. N., Kurtz T. G., Markov Processes, Wiley Series in Probability and Mathematical Statistics, John Wiley & Sons Inc., New York, 1986. | MR 838085 | Zbl 0592.60049

[12] Freidlin M., The Dirichlet Problem for an Equation With Periodic Coefficients Depending on a Small Parameter, Teor. Veroyatnost. i Primenen. 9 (1964) 133-139. | MR 163062 | Zbl 0138.11602

[13] Hairer M., Pardoux E., Homogenization of Periodic Linear Degenerate PDEs, available on, arXiv: math/0702304v1. | MR 2473263 | Zbl 1167.60015

[14] Heron B., Mossino J., H-Convergence and Regular Limits for Stratified Media With Low and High Conductivities, Appl. Anal. 57 (3-4) (1995) 271-308. | MR 1384376 | Zbl 0833.35010

[15] Huang Y., Su N., Zhang X., Homogenization of Degenerate Quasilinear Parabolic Equations With Periodic Structure, Asymptotic Anal. 48 (2006) 77-89. | MR 2233379 | Zbl 1157.35313

[16] Jakobsen E. R., Karlsen K. H., Continuous Dependence Estimates for Viscosity Solutions of Fully Nonlinear Degenerate Parabolic Equations, J. Differential Equations 183 (2002) 497-525. | MR 1919788 | Zbl 1086.35061

[17] A. Lejay, Méthodes probabilistes pour l'homogénéisation des opérateurs sous forme divergence, Thèse de l'université de Provence, 2000.

[18] Meyer P. A., Zheng W. A., Tightness Criteria for Laws of Semartingales, Ann. Inst. Henri Poincaré 20 (1984) 353-372. | Numdam | MR 771895 | Zbl 0551.60046

[19] Pardoux E., Homogenization of a Linear and Semilinear Second Order Parabolic PDEs With Periodic Coefficients: a Probabilistic Approach, J. Funct. Anal. 167 (1999) 498-520. | MR 1716206 | Zbl 0935.35010

[20] Pardoux E., BSDEs, Weak Convergence and Homogenization of Semilinear PDEs, in: Clarke F. H., Stern R. J. (Eds.), Nonlinear Analysis, Differential Equations and Control, Kluwer Acad. Publ., 1999, pp. 503-549. | MR 1695013 | Zbl 0959.60049

[21] Pardoux E., Backward Stochastic Differential Equations and Viscosity Solutions of Systems of Semilinear Parabolic and Elliptic PDEs of Second Order, in: Decreusefond L., Gjerde J., Oksendal B., Üstünel A. S. (Eds.), Stochastic Analysis and Related Topics VI: the Geilo Workshop, 1996, Birkhäuser, 1998, pp. 79-127. | MR 1652339 | Zbl 0893.60036

[22] Pardoux E., Peng S., Backward Stochastic Differential Equations and Quasilinear Parabolic Partial Differential Equations, in: Rozovskii B. L., Sowers R. S. (Eds.), Stochastic Partial Differential Equations and Their Applications, Lecture Notes in Control & Information Sciences, vol. 176, Springer, Berlin, 1992, pp. 200-217. | MR 1176785 | Zbl 0766.60079

[23] Pardoux E., Veretennikov A. Yu., Averaging of Backward Stochastic Differential Equations With Application to Semilinear PDE's, Stochastics Stochastic Rep. 60 (1997) 255-270. | MR 1467720 | Zbl 0891.60053

[24] Pardoux E., Veretennikov A. Yu., On Poisson Equation and Diffusion Approximation 1, Ann. Probab. 29 (2001) 1061-1085. | MR 1872736 | Zbl 1029.60053

[25] Pardoux E., Yu Veretennikov A., On Poisson Equation and Diffusion Approximation 3, Ann. Probab. 33 (2005). | MR 2135314 | Zbl 1071.60022

[26] Paronetto F., Serra Cassano F., On the Convergence of a Class of Degenerate Parabolic Equations, J. Math. Pures Appl. 77 (1998) 851-878. | MR 1656776 | Zbl 0920.35082

[27] Paronetto F., Homogenization of a Class of Degenerate Parabolic Equations, Asymptotic Anal. 21 (1999) 275-302. | MR 1728026 | Zbl 0945.35009

[28] Paronetto F., Homogenization of Degenerate Elliptic-Parabolic Equations, Asymptotic Anal. 37 (2004) 21-56. | MR 2035361 | Zbl 1052.35025

[29] Rhodes R., On Homogenization of Space-Time Dependent and Degenerate Random Flows, Stochastic Process. Appl. 117 (2007) 1561-1585. | MR 2353040 | Zbl 1127.60027

[30] R. Rhodes, Homogenization of locally ergodic diffusions with possibly degenerate diffusion matrix, Ann. Inst. H. Poincaré Probab. Statist., in press.