Superposition rules and stochastic Lie–Scheffers systems

Joan-Andreu Lázaro-Camí; Juan-Pablo Ortega

Annales de l'I.H.P. Probabilités et statistiques (2009)

  • Volume: 45, Issue: 4, page 910-931
  • ISSN: 0246-0203

Abstract

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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.

How to cite

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Lázaro-Camí, Joan-Andreu, and Ortega, Juan-Pablo. "Superposition rules and stochastic Lie–Scheffers systems." Annales de l'I.H.P. Probabilités et statistiques 45.4 (2009): 910-931. <http://eudml.org/doc/78061>.

@article{Lázaro2009,
abstract = {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.},
author = {Lázaro-Camí, Joan-Andreu, Ortega, Juan-Pablo},
journal = {Annales de l'I.H.P. Probabilités et statistiques},
keywords = {Lie–Scheffers system; superposition rules; stochastic differential equations; Wei–Norman method; Lie-Scheffers system; Wei-Norman method},
language = {eng},
number = {4},
pages = {910-931},
publisher = {Gauthier-Villars},
title = {Superposition rules and stochastic Lie–Scheffers systems},
url = {http://eudml.org/doc/78061},
volume = {45},
year = {2009},
}

TY - JOUR
AU - Lázaro-Camí, Joan-Andreu
AU - Ortega, Juan-Pablo
TI - Superposition rules and stochastic Lie–Scheffers systems
JO - Annales de l'I.H.P. Probabilités et statistiques
PY - 2009
PB - Gauthier-Villars
VL - 45
IS - 4
SP - 910
EP - 931
AB - 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.
LA - eng
KW - Lie–Scheffers system; superposition rules; stochastic differential equations; Wei–Norman method; Lie-Scheffers system; Wei-Norman method
UR - http://eudml.org/doc/78061
ER -

References

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  1. [1] F. Baudoin. An Introduction to the Geometry of Stochastic Flows. Imperial College Press, London, 2004. Zbl1085.60002MR2154760
  2. [2] G. Ben Arous. Flots et series de Taylor stochastiques. Probab. Theory Related Fields 81 (1989) 29–77. Zbl0639.60062MR981567
  3. [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. Zbl1221.34025MR1810256
  4. [4] J. F. Cariñena, J. Grabowski and G. Marmo. Superposition rules, Lie theorem, and partial differential equations. Rep. Math. Phys. 60 (2007) 237–258. Zbl1153.34004MR2374820
  5. [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. Zbl0928.34025
  6. [6] J. F. Cariñena and A. Ramos. A new geometric approach to Lie systems and physical applications. Acta Appl. Math. 70 (2002) 43–69. Zbl1003.34007MR1892375
  7. [7] F. Castell. Asymptotic expansion of stochastic flows. Probab. Theory Related Fields 96 (1993) 225–239. Zbl0794.60054MR1227033
  8. [8] P. Dazord. Feuilletages à singularités. Nederl. Akad. Wetensch. Indag. Math. 47 (1985) 21–39. Zbl0584.57016MR783003
  9. [9] K. D. Elworthy. Stochastic Differential Equations on Manifolds. London Mathematical Society Lecture Notes Series 70. Cambridge Univ. Press, 1982. Zbl0514.58001MR675100
  10. [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. Zbl0942.58004MR1735806
  11. [11] M. Émery. Stochastic Calculus in Manifolds. Springer, Berlin, 1989. Zbl0697.60060MR1030543
  12. [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. Zbl0980.60085MR1837291
  13. [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. Zbl0609.60009MR942031
  14. [14] S. Helgason. Differential Geometry, Lie Groups and Symmetric Spaces. Pure and Applied Mathematics 80. Academic Press, New York, 1978. Zbl0451.53038MR514561
  15. [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. Zbl0766.60069MR1232020
  16. [16] S. Kobayashi and K. Nomizu. Foundations of Differential Geometry II. Tracts in Mathematics 15. Wiley, New York, 1969. Zbl0175.48504
  17. [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. Zbl0438.60047MR580134
  18. [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. Zbl1187.60045MR2502472
  19. [19] M. Liao. Lévy Processes in Lie Groups. Cambridge Tracts in Mathematics 162. Cambridge Univ. Press, 2004. Zbl1076.60004MR2060091
  20. [20] S. Lie. Vorlesungen über Continuierliche Gruppen mit Geometrischen und Anderen Andwendungen. Teubner, Leipzig, 1893. (G. Scheffers.) Zbl0248.22010MR392458JFM25.0623.02
  21. [21] T. J. Lyons. Differential equations driven by rough signals. Rev. Mat. Iberoamericana 14 (1998) 215–310. Zbl0923.34056MR1654527
  22. [22] Malliavin, P.Géométrie Différentielle Stochastique. Séminaire de Mathématiques Supérieures 64. Presses de l’Université de Montréal, 1978. Zbl0393.60062
  23. [23] R. Palais. A global formulation of the Lie theory on transformation groups. Mem. Amer. Math. Soc. 22 (1957) 95–97. Zbl0178.26502MR121424
  24. [24] P. Stefan. Accessibility and foliations with singularities. Bull. Amer. Math. Soc. 80 (1974) 1142–1145. Zbl0293.57015MR353362
  25. [25] P. Stefan. Accessible sets, orbits and foliations with singularities. Proc. Lond. Math. Soc. 29 (1974) 699–713. Zbl0342.57015MR362395
  26. [26] H. Sussman. Orbits of families of vector fields and integrability of distributions. Trans. Amer. Math. Soc. 180 (1973) 171–188. Zbl0274.58002MR321133
  27. [27] J. Wei and E. Norman. Lie algebraic solution of linear differential equations. J. Math. Phys. 4 (1963) 575–581. Zbl0133.34202MR149053
  28. [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. Zbl0119.07202MR160009

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