Potential Analysis

, Volume 38, Issue 1, pp 169–205 | Cite as

Iteration of the Lent Particle Method for Existence of Smooth Densities of Poisson Functionals

  • Nicolas Bouleau
  • Laurent Denis


In previous works (Bouleau and Denis, J Funct Anal 257:1144–1174, 2009, Probab Theory Relat Fields, 2011) we have introduced a new method called the lent particle method which is an efficient tool to establish existence of densities for Poisson functionals. We now go further and iterate this method in order to prove smoothness of densities. More precisely, we construct Sobolev spaces of any order and prove a Malliavin-type criterion of existence of smooth density. We apply this approach to SDE’s driven by Poisson random measures and also present some non-trivial examples to which our method applies.


Stochastic differential equation Poisson functional Dirichlet Form Energy image density Lévy processes Gradient Carré du champ 

Mathematics Subject Classifications (2010)

Primary 60G57 60H05; Secondary 60J45 60G51 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Ané, C., et al.: Sur les inégalités de Sobolev logarithmiques. Panoramas et Synthèses, SMF (2000)Google Scholar
  2. 2.
    Bally, V., Clément, E.: Integration by parts formula and applications to equations with jumps. Probab. Theory Relat. Fields 151(3–4), 613–657 (2011)zbMATHCrossRefGoogle Scholar
  3. 3.
    Bakry, D.: Transformations de Riesz pour les semigroupes symétriques. Sém Proba XIX, LNM 1123. Springer (1985)Google Scholar
  4. 4.
    Bakry, D., Emery, M.: Diffusions hypercontractives, p. 177. Sém. Strasbourg XIX. Springer, New York (1985)Google Scholar
  5. 5.
    Bichteler, K., Gravereaux, J.-B., Jacod, J.: Malliavin Calculus for Processes with Jumps. Gordon and Breach Science Publishers (1987)Google Scholar
  6. 6.
    Bichteler, K., Jacod, J.: Calcul de Malliavin pour les diffusions avec sauts, existence d’une densité dans le cas uni-dimensionnel. In: Séminaire de Probabilités XVII. Lect. Notes in Math., vol. 986, pp. 132–157. Springer, New York (1983)Google Scholar
  7. 7.
    Bingham, N.H., Goldie, C.M., Teugels, J.L.: Regular Variation. Cambridge University Press, Cambridge, MA (1987)zbMATHGoogle Scholar
  8. 8.
    Bismut, J.-M.: Mécanique Aléatoire. Lect. Notes in Math., vol. 866. Springer, New York (1981)zbMATHGoogle Scholar
  9. 9.
    Bouleau, N.: Error calculus and regularity of Poisson functionals: the lent particle method. C. R. Acad. Sc. Paris, Mathématiques 346(13–14), 779–782 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Bouleau, N., Hirsch, F.: Dirichlet Forms and Analysis on Wiener Space. De Gruyter (1991)Google Scholar
  11. 11.
    Bouleau, N., Denis, L.: Energy image density property and the lent particle method for Poisson measures. J. Funct. Anal. 257, 1144–1174 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Bouleau, N., Denis, L.: Application of the lent particle method to Poisson-driven SDEs. Probab. Theory Relat. Fields 151(3–4), 403–433 (2011)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Coquio, A.: Formes de Dirichlet sur l’espace canonique de Poisson et application aux équations différentielles stochastiques. Ann. Inst. Henri Poincaré 19(1), 1–36 (1993)MathSciNetGoogle Scholar
  14. 14.
    Denis, L.: Quasi-sure analysis for the Euler approximation and the flow related to an S.D.E. In: Dirichlet Forms and Stochastic Processes, pp. 103–112. De Gruyter (1995)Google Scholar
  15. 15.
    Ikeda, N., Watanabe, S.: Stochastic Differential Equation and Diffusion Processes. North-Holland, Koshanda (1981)Google Scholar
  16. 16.
    Jacod, J.: Calcul Stochastique et Problèmes de Martingales. Lect. Notes in Math., vol. 714. Springer, New York (1979)zbMATHGoogle Scholar
  17. 17.
    Kunita, H.: Stochastic Flows and Stochastic Differential Equations. Cambridge University Press, Cambridge, MA (1990)zbMATHGoogle Scholar
  18. 18.
    Latala, R.: Estimates of moments and tails of Gaussian chaoses. Ann. Probab. 34(6), 2315–2331 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Léandre, R.: Régularité de processus de sauts dégénérés (I), (II). Ann. Inst. Henri Poincaré 21, 125–146 (1985); 24, 209–236 (1988)Google Scholar
  20. 20.
    Léandre, R.: Regularity of a degenerated convolution semi-group without to use the Poisson process. In: Luo, A. (ed.) Proceedings of the Non linear Science and Complexity, pp. 311–320, Springer, Porto, Portugal (2010)Google Scholar
  21. 21.
    Ledoux, M., Talagrand, M.: Probability in Banach Spaces. Springer, New York (1991)zbMATHGoogle Scholar
  22. 22.
    Meyer, P.A.: Transformations de Riesz pour les lois gaussiennens. Sém. Strasbourg XVIII, p. 179. Springer, New York (1984)Google Scholar
  23. 23.
    Picard, J.: On the existence of smooth densities for jump processes. Probab. Theory Relat. Fields 105, 481–511 (1996)MathSciNetzbMATHCrossRefGoogle Scholar
  24. 24.
    Pisier, G.: Les inégalités de Khintchine-Kahane, d’après C. Borell. Séminaire Analyse fonctionnelle (dit “Maurey-Schwartz”), exp. 7, (1977–1978). (Available on
  25. 25.
    Stein, E.M.: Some results in harmonic analysis in ℝn for n→ ∞. Bull. Am. Math. Soc. 9, 71–73 (1983)zbMATHCrossRefGoogle Scholar
  26. 26.
    Wu, L.: Inégalité de Sobolev sur l’espace de Poisson. Sém. Proba. Strasbourg XXI, p. 114. Springer, New York (1987)Google Scholar

Copyright information

© Springer Science+Business Media B.V. 2011

Authors and Affiliations

  1. 1.Ecole des PontsParisTech, Paris-EstMarne-La-Vallée Cedex 2France
  2. 2.Equipe Analyse et ProbabilitésUniversité d’Evry-Val-d’EssonneEVRY CedexFrance

Personalised recommendations