Superposition rules and stochastic Lie-Scheffers systems
Lázaro-Camí, Joan-Andreu ; Ortega, Juan-Pablo
Annales de l'I.H.P. Probabilités et statistiques, Tome 45 (2009), p. 910-931 / Harvested from Numdam
Access to full text
Full (PDF)
Full (DJVU)

Ce papier contient une généralisation du Théorème de Lie-Scheffers aux équations différentielles stochastiques. Ce résultat caractérise l'existence de règles de superposition non linéaires pour la solution générale de ces équations, en termes des propriétés d'involution de la distribution engendrée par les champs vecteurs qui les définissent. Dans le cas particulier des systèmes déterministes, notre théorème principal améliore certains aspects du théorème de Lie-Scheffers traditionnel. Nous montrons que l'analogue stochastique des systèmes de Lie-Scheffers classiques peuvent être réduits à l'étude des systèmes de Lie-Scheffers stochastiques à valeurs dans un groupe de Lie; ces systèmes, ainsi que ceux qui prennent des valeurs dans des espaces homogènes sont étudiés en détail. Les développements de ce papier sont illustrés avec plusieurs exemples.

This paper proves a version for stochastic differential equations of the Lie-Scheffers theorem. This result characterizes the existence of nonlinear superposition rules for the general solution of those equations in terms of the involution properties of the distribution generated by the vector fields that define it. When stated in the particular case of standard deterministic systems, our main theorem improves various aspects of the classical Lie-Scheffers result. We show that the stochastic analog of the classical Lie-Scheffers systems can be reduced to the study of Lie group valued stochastic Lie-Scheffers systems; those systems, as well as those taking values in homogeneous spaces are studied in detail. The developments of the paper are illustrated with several examples.

Publié le : 2009-01-01
DOI : https://doi.org/10.1214/08-AIHP189
Classification:  60H10,  34F05 
@article{AIHPB_2009__45_4_910_0,
     author = {L\'azaro-Cam\'\i , Joan-Andreu and Ortega, Juan-Pablo},
     title = {Superposition rules and stochastic Lie-Scheffers systems},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     volume = {45},
     year = {2009},
     pages = {910-931},
     doi = {10.1214/08-AIHP189},
     mrnumber = {2572157},
     zbl = {1196.60107},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AIHPB_2009__45_4_910_0}
}

Lázaro-Camí, Joan-Andreu; Ortega, Juan-Pablo. Superposition rules and stochastic Lie-Scheffers systems. Annales de l'I.H.P. Probabilités et statistiques, Tome 45 (2009) pp. 910-931. doi : 10.1214/08-AIHP189. http://gdmltest.u-ga.fr/item/AIHPB_2009__45_4_910_0/

[1] F. Baudoin. An Introduction to the Geometry of Stochastic Flows. Imperial College Press, London, 2004. | MR 2154760 | Zbl 1085.60002

[2] G. Ben Arous. Flots et series de Taylor stochastiques. Probab. Theory Related Fields 81 (1989) 29-77. | MR 981567 | Zbl 0639.60062

[3] J. F. Cariñena, J. Grabowski and G. Marmo. Lie-Scheffers Systems: A Geometric Approach. Napoli Series on Physics and Astrophysics 3. Bibliopolis, Naples, 2000. | MR 1810256 | Zbl pre01596788

[4] J. F. Cariñena, J. Grabowski and G. Marmo. Superposition rules, Lie theorem, and partial differential equations. Rep. Math. Phys. 60 (2007) 237-258. | MR 2374820 | Zbl 1153.34004

[5] J. F. Cariñena, G. Marmo and J. Nasarre. The nonlinear superposition principle and the Wei-Norman method. Internat. J. Modern Phys. A 13 (1998) 3601-3627. | Zbl 0928.34025

[6] J. F. Cariñena and A. Ramos. A new geometric approach to Lie systems and physical applications. Acta Appl. Math. 70 (2002) 43-69. | MR 1892375 | Zbl 1003.34007

[7] F. Castell. Asymptotic expansion of stochastic flows. Probab. Theory Related Fields 96 (1993) 225-239. | MR 1227033 | Zbl 0794.60054

[8] P. Dazord. Feuilletages à singularités. Nederl. Akad. Wetensch. Indag. Math. 47 (1985) 21-39. | MR 783003 | Zbl 0584.57016

[9] K. D. Elworthy. Stochastic Differential Equations on Manifolds. London Mathematical Society Lecture Notes Series 70. Cambridge Univ. Press, 1982. | MR 675100 | Zbl 0514.58001

[10] K. D. Elworthy, Y. Le Jan and X.-M. Li. On the Geometry of Diffusion Operators and Stochastic Flows. Lecture Notes in Mathematics 1720. Springer, Berlin, 1999. | MR 1735806 | Zbl 0942.58004

[11] M. Émery. Stochastic Calculus in Manifolds. Springer, Berlin, 1989. | MR 1030543 | Zbl 0697.60060

[12] A. Estrade and M. Pontier. Backward stochastic differential equations in a Lie group. In Séminaire de probabilités (Strasbourg), XXXV. 241-259. Lecture Notes in Math. 1755. Springer, Berlin, 2001. | Numdam | MR 1837291 | Zbl 0980.60085

[13] M. Hakim-Dowek and D. Lepingle. L'exponentielle stochastique des groupes de Lie. In Séminaire de Probabilités (Strasbourg), XX 352-374. Lecture Notes in Math. 1204. Springer, Berlin, 1986. | Numdam | MR 942031 | Zbl 0609.60009

[14] S. Helgason. Differential Geometry, Lie Groups and Symmetric Spaces. Pure and Applied Mathematics 80. Academic Press, New York, 1978. | MR 514561 | Zbl 0451.53038

[15] Y.-Z. Hu. Série de Taylor stochastique et formule de Campbell-Hausdorff, d'après Ben Arous. In Séminaire de Probabiliés (Strasbourg), XXVI 579-586. Lecture Notes in Math. 1526. Springer, Berlin, 1992. | Numdam | MR 1232020 | Zbl 0766.60069

[16] S. Kobayashi and K. Nomizu. Foundations of Differential Geometry II. Tracts in Mathematics 15. Wiley, New York, 1969. | Zbl 0175.48504

[17] H. Kunita. On the representation of solutions of stochastic differential equations. In Séminaire de Probabilités (Strasbourg), XIV 282-304. Lecture Notes in Math. 784. Springer, Berlin, 1980. | Numdam | MR 580134 | Zbl 0438.60047

[18] J.-A. Lázaro-Camí and J.-P. Ortega. Reduction, reconstruction, and skew-product decomposition of symmetric stochastic differential equations. Stoch. Dyn. (2009). To appear. Available at http://arxiv.org/abs/0705.3156. | MR 2502472 | Zbl pre05548747

[19] M. Liao. Lévy Processes in Lie Groups. Cambridge Tracts in Mathematics 162. Cambridge Univ. Press, 2004. | MR 2060091 | Zbl 1076.60004

[20] S. Lie. Vorlesungen über Continuierliche Gruppen mit Geometrischen und Anderen Andwendungen. Teubner, Leipzig, 1893. (G. Scheffers.) | JFM 25.0623.02 | MR 392458

[21] T. J. Lyons. Differential equations driven by rough signals. Rev. Mat. Iberoamericana 14 (1998) 215-310. | MR 1654527 | Zbl 0923.34056

[22] Malliavin, P. Géométrie Différentielle Stochastique. Séminaire de Mathématiques Supérieures 64. Presses de l'Université de Montréal, 1978. | Zbl 0393.60062

[23] R. Palais. A global formulation of the Lie theory on transformation groups. Mem. Amer. Math. Soc. 22 (1957) 95-97. | MR 121424 | Zbl 0178.26502

[24] P. Stefan. Accessibility and foliations with singularities. Bull. Amer. Math. Soc. 80 (1974) 1142-1145. | MR 353362 | Zbl 0293.57015

[25] P. Stefan. Accessible sets, orbits and foliations with singularities. Proc. Lond. Math. Soc. 29 (1974) 699-713. | MR 362395 | Zbl 0342.57015

[26] H. Sussman. Orbits of families of vector fields and integrability of distributions. Trans. Amer. Math. Soc. 180 (1973) 171-188. | MR 321133 | Zbl 0274.58002

[27] J. Wei and E. Norman. Lie algebraic solution of linear differential equations. J. Math. Phys. 4 (1963) 575-581. | MR 149053 | Zbl 0133.34202

[28] J. Wei and E. Norman. On global representations of the solutions of linear differential equations as a product of exponentials. Proc. Amer. Math. Soc. 15 (1964) 327-334. | MR 160009 | Zbl 0119.07202