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) no. 4, pp. 910-931.

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.

Classification : 60H10, 34F05
Mots clés : Lie-Scheffers system, superposition rules, stochastic differential equations, Wei-Norman method

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

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

[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

[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 | Zbl

[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

[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 | Zbl

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

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

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

[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 | Zbl

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

[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 | Zbl

[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 | Zbl

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

[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 | Zbl

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

[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 | Zbl

[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

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

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

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

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

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

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

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

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

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

[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 | Zbl