Skip to main content
Log in

Malliavin calculus of Bismut type for an operator of order four on a Lie group

  • Published:
Journal of Pseudo-Differential Operators and Applications Aims and scope Submit manuscript

Abstract

We give an adaptation of the Malliavin calculus of Bismut type for a semi-group generated by a four order operator on a Lie group. In such a case, the semi-group does not preserve the positivity.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. Albeverio, S., Hoegh-Krohn, R.: Dirichlet forms and diffusion processes on rigged Hilbert spaces. Z. Wahr. Verw. Geb. 40, 1–57 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  2. Berezanskii, Y.: The selfadjointness of elliptic operators with infnite number of variables. Ukr. J. Math. 27, 729–742 (1975)

    Google Scholar 

  3. Bismut, J.M.: Martingales, the Malliavin calculus and hypoellipticity under general Hoermander’s conditions. Z. Wahr. Verw. Geb. 63, 469–505 (1981)

    Article  MATH  Google Scholar 

  4. Chazarain, J., Piriou, A.: Introduction a la théorie des équations aux dérivées partielles linéaires. Gauthier-Villars, Paris (1981)

    MATH  Google Scholar 

  5. Davies, E.B.: \(L^p\) spectral theory of higher-order elliptic differential equations. Bull. Lond. Math. Soc. 29, 513–546 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  6. Dieudonné, J.: Eléments d’analyse, vol. VII. Gauthiers-Villars, Paris (1977)

    Google Scholar 

  7. Gross, L.: Potential theory on a Hilbert space. J. Funct. Anal. 1, 123–181 (1966)

    Article  MathSciNet  MATH  Google Scholar 

  8. Hida, T.: Analysis of Brownian Functionals. Carleton Mathematical Lecture Notes, vol. 13. Carleton University Press, Ottawa (1975)

  9. Hoermander, L.: The Analysis of Linear Partial Operators IV. Springer, Berlin (1984)

    Google Scholar 

  10. Hoermander, L.: The Analysis of Linear Partial Operators III. Springer, Berlin (1984)

    Google Scholar 

  11. Léandre, R.: A Girsanov formula associated to a big order pseudodifferential operator. In: Festchrift in honour of G. N’Guérékata, B. Anglade et al. (eds.) Cubo: a Mathematical Journal, vol. 15, pp. 113–119 (2013)

  12. Léandre, R.: Applications of the Malliavin calculus of Bismut type without probability”. In: Madureira, A.M., W.S.E.A.S., CD. (eds.) 6th International Conference on Simulation, Modeling and Optimization, WSEAS Transactions on Mathematics, vol. 5, pp. 559–564, 1205–1210 (2006)

  13. Léandre, R.: Applications quantitatives et qualitatives du Calcul de Malliavin. In: Séminaire Franco-Japonais, S., Métivier, M., Watanabe, S. (eds.) Lectures Notes Mathematics, pp. 109–134. Springer, Berlin (1988). English translation: In: Geometry of Random Motion. Durrett, R., Pinsky, M. (eds.) Contemporary Mathematics 73, pp. 173–196. A.M.S., Providence (1988)

  14. Léandre, R.: Large deviation estimates for a non-Markovian Lévy generator of big order. In: Vagenas, E., et al. (eds.) 4th International Conference on Mathematical Modern Physical Sciences. Journal of Physics: Conference Series, vol. 633, pp. 012085 (2015)

  15. Léandre, R.: Malliavin of Bismut type without probability. In: Festchrift in Honour of K. Sinha, V.S. Sunder (eds.) Proceedings of the Indian Academy of Science and Mathematical Science, vol. 116, pp. 507–518 (2006)

  16. Léandre, R.: Perturbation of the Malliavin calculus of Bismut type of large order. In: Gazeau, J.P., et al. (eds.) To appear in XXXI International Congress Group of Methods in Physics

  17. Léandre, R.: The Itô–Stratonovitch formula for an operator of order four. In: Festchrift in honour of A. Mikherjea, P. Feinsilver, S. Mohammed. Budzhan, G., et al. (eds.) Contemporary Mathematics, vol. 668, pp. 164–168 (2016)

  18. Léandre, R.: The stochastic flow theorem for an operator of order four. In: Nielsen, F., et al. (eds.) Geometry of Science Information 2013, pp. 497–502. L.N.C.S. 8085, Springer, Berlin (2013)

  19. Léandre, R.: Varadhan estimates for an operator of order four on a Lie group. In: Viedma, E.D., et al. (eds.) Control, Decision and Information Technologies, IEEE, pp. 15. Los-Alamitos, IEEE-Xplore (2016)

  20. Léandre, R.: Wentzel–Freidlin estimates for an operator of order four. In: Akhgar, B., et al. (eds.) 2014 International Conference on Computer Science and Computer Intel, pp. 360–364. IEEE Computer Society. IEEE-Xplore, Los-Alamitos (2014)

  21. Léandre, R.: Malliavin calculus of Bismut type in semi-group theory. Far East J. Math. Sci. 30,n, 1–26 (2008)

    MathSciNet  MATH  Google Scholar 

  22. Léandre, R.: Stochastic analysis for a non-Markovian generator: an introduction. Russ. J. Math. Phys. 22, 39–52 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  23. Malliavin, P.: Stochastic calculus of variations and hypoelliptic operators. In: Itô, K. (ed.) Proceedings of International Symposium Stochastic Differential Equations, Kyoto, pp. 195–263. Kinokuyina, Tokyo (1978)

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Rémi Léandre.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Léandre, R. Malliavin calculus of Bismut type for an operator of order four on a Lie group. J. Pseudo-Differ. Oper. Appl. 9, 419–430 (2018). https://doi.org/10.1007/s11868-017-0190-3

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11868-017-0190-3

Keywords

Navigation