Skip to main content
SpringerLink
Log in

Lévy Processes and Infinitely Divisible Measures in the Dual of a Nuclear Space

  • Published:
Journal of Theoretical Probability Aims and scope Submit manuscript

Abstract

Let \(\Phi \) be a nuclear space and let \(\Phi '_{\beta }\) denote its strong dual. In this work, we prove the existence of càdlàg versions, the Lévy–Itô decomposition and the Lévy–Khintchine formula for \(\Phi '_{\beta }\)-valued Lévy processes. Moreover, we give a characterization for Lévy measures on \(\Phi '_{\beta }\) and provide conditions for the existence of regular versions to cylindrical Lévy processes in \(\Phi '\). Furthermore, under the assumption that \(\Phi \) is a barrelled nuclear space we establish a one-to-one correspondence between infinitely divisible measures on \(\Phi '_{\beta }\) and Lévy processes in \(\Phi '_{\beta }\). Finally, we prove the Lévy–Khintchine formula for infinitely divisible measures on \(\Phi '_{\beta }\).

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

Access this article

Log in via an institution

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

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Applebaum, D.: Lévy Processes and Stochastic Calculus, 2nd edn. Cambridge Studies in Advanced Mathematics, Cambridge (2009)

    Book  Google Scholar 

  2. Applebaum, D., Riedle, M.: Cylindrical Lévy processes in Banach spaces. Proc. Lond. Math. Soc. 101(3), 697–726 (2010)

    Article  MathSciNet  Google Scholar 

  3. Badrikian, A.: Séminaire Sur les Fonctions Aléatoires Linéaires et les Mesures Cylindriques. Lecture Notes in Mathematics, vol. 139. Springer, Berlin (1970)

    Book  Google Scholar 

  4. Baumgartner, F.: Lévy processes with values in locally convex Suslin spaces. arXiv:1510.00538

  5. Bogachev, V.I.: Measure Theory, vol. I–II. Springer, Berlin (2007)

    Book  Google Scholar 

  6. Bojdecki, T., Jakubowski, J.: Stochastic integration for inhomogeneous wiener process in the dual of a nuclear space. J. Multivar. Anal. 34, 185–210 (1990)

    Article  MathSciNet  Google Scholar 

  7. Bretagnolle, J.L.: Processus à accroissements indépendants. In: Ecole d’Été de Probabilités: Processus Stochastiques, Lecture Notes in Mathematics, vol. 307, pp. 1–26. Springer, Berlin (1973)

  8. Chung, D.M., Rajput, B.S., Tortrat, A.: Semistable laws on topological vector spaces. Z. Wahrsch. Verw. Gebiete 60(2), 209–218 (1982)

    Article  MathSciNet  Google Scholar 

  9. Dalecky, Y.L., Fomin, S.V.: Measure and Differential Equations in Infinite-Dimensional Space, Mathematics and Its Applications, vol. 76. Springer, Berlin (1991)

    Google Scholar 

  10. Dettweiler, E.: Grenzwertsätze für Wahrscheinlichkeitsmaße auf Badrikianschen Räumen. Z. Wahrsch. Verw. Gebiete 34, 285–311 (1976)

    Article  Google Scholar 

  11. Dettweiler, E.: Infinitely divisible measures on the cone of an ordered locally convex vector space. Ann. Sci. Univ. Clermont 14(61), 11–17 (1976)

    MathSciNet  MATH  Google Scholar 

  12. Dineen, S.: Complex Analysis in Locally Convex Spaces, North-Holland Mathematics Studies, vol. 57. Notas de Mat., No. 83, North-Holland (1981)

  13. Fernique, X.: Séries de distributions aléatoires indépendantes. I: Généralités sur les distributions aléatoires. Sémin. Probab. (Strasbg.) 1, 54–64 (1967)

    Google Scholar 

  14. Fernique, X.: Lois indéfinement divisibles sur l’space des distributions. Invent. Math. 3, 282–292 (1967)

    Article  MathSciNet  Google Scholar 

  15. Fonseca-Mora, C.A.: Stochastic analysis with Lévy noise in the dual of a nuclear space. Ph.D. Thesis, The University of Sheffield (2015)

  16. Fonseca-Mora, C.A.: Existence of continuous and càdlàg versions for cylindrical processes in the dual of a nuclear space. J. Theor. Probab. 31(2), 867–894 (2018)

    Article  MathSciNet  Google Scholar 

  17. Fonseca-Mora, C.A.: Stochastic integration and stochastic PDEs driven by jumps on the dual of a nuclear space. Stoch PDE: Anal. Comput. 6(4), 618–689 (2018)

    Article  MathSciNet  Google Scholar 

  18. Fonseca-Mora, C.A.: Tightness and weak convergence of probabilities on the Skorokhod space on the dual of a nuclear space and applications. Studia Math. arXiv:1806.10231

  19. Fonseca-Mora, C.A.: Semimartingales on duals of nuclear spaces. arXiv:1902.03981

  20. Hogbe-Nlend, H., Moscatelli, V.: Nuclear and Conuclear Spaces, North-Holland Mathematical Studies, vol. 52. Notas de Mat., No. 79, North-Holland (1981)

  21. Ikeda, N., Watanabe, S.: Stochastic Differential Equations and Diffusion Processes. North-Holland Mathematical Library, 2nd edn. North-Holland/Kodansha, Amsterdam (1989)

    MATH  Google Scholar 

  22. Itô, K.: Foundations of Stochastic Equations in Infinite Dimensional Spaces. Series in Mathematics. SIAM, Philadelphia (1984)

    Book  Google Scholar 

  23. Jarchow, H.: Locally Convex Spaces, Mathematische Leitfäden. Springer, Berlin (1981)

    Book  Google Scholar 

  24. Kallenberg, O.: Foundations of Modern Probability, Probability and Its Applications, 2nd edn. Springer, New York (2002)

    Book  Google Scholar 

  25. Kallianpur, G., Xiong, J.: Stochastic Differential Equations in Infinite Dimensional Spaces. Lecture Notes–Monograph Series. Institute of Mathematical Statistics, Hayward, CA (1995)

  26. Medvegyev, P.: Stochastic Integration Theory. Oxford Graduate Text in Mathematics. Oxford University Press, Oxford (2007)

    MATH  Google Scholar 

  27. Narici, L., Beckenstein, E.: Topological Vector Spaces. Pure and Applied Mathematics, 2nd edn. CRC Press, New York (2011)

    MATH  Google Scholar 

  28. Parthasarathy, K.R.: Probability Measures on Metric Spaces. New York Academic Press, New York (1967)

    Book  Google Scholar 

  29. Pérez-Abreu, V., Rocha-Arteaga, A., Tudor, C.: Cone-additive processes in duals of nuclear Féchet spaces. Random Oper. Stoch. Equ. 13(4), 353–368 (2005)

    Article  Google Scholar 

  30. Pérez-Abreu, V., Rosiński, J.: Representation of infinitely divisible distributions on cones. J. Theor. Probab. 20(3), 535–544 (2007)

    Article  MathSciNet  Google Scholar 

  31. Pietsch, A.: Nuclear Locally Convex Spaces, Ergebnisse der Mathematikund ihrer Grenzgebiete. Springer, Berlin (1972)

    Book  Google Scholar 

  32. Rajput, B.S.: On the support of symmetric infinitely divisible and stable probability measures on LCTVS. Proc. Am. Math. Soc. 66(2), 331–334 (1977)

    Article  MathSciNet  Google Scholar 

  33. Rao, M.M.: Stochastic Processes: General Theory. Mathematics and Its Applications. Springer, New York (1995)

    Book  Google Scholar 

  34. Riedle, M., van Gaans, O.: Stochastic integration for Lévy processes with values in banach spaces. Stoch. Process. Appl. 119, 1952–1974 (2009)

    Article  Google Scholar 

  35. Sato, K.-I.: Lévy Processes and Infinitely Divisible Distributions. Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge (1999)

    MATH  Google Scholar 

  36. Schaefer, H.: Topological Vector Spaces. Graduate Texts in Mathematics, 2nd edn. Springer, New York (1999)

    Book  Google Scholar 

  37. Schwartz, L.: Radon Measures on Arbitrary Topological Spaces and Cylindrical Measures. Tata Institute of Fundamental Research Studies in Mathematics. Oxford University Press, Oxford (1973)

    MATH  Google Scholar 

  38. Schwartz, L.: Processus de Markov et désintégrations régulières. Ann. Inst. Fourier (Grenoble) 27(3), 211–277 (1977). xi

    Article  MathSciNet  Google Scholar 

  39. Siebert, E.: Einbettung unendlich teilbarer wahrscheinlichkeitsmaße auf topologischen gruppen. Z. Wahrscheinlichkeitstheorie Verw. Geb. 28, 227–247 (1974)

    Article  Google Scholar 

  40. Siebert, E.: Convergence and convolutions of probability measures on a topological group. Ann. Probab. 4(3), 433–443 (1976)

    Article  MathSciNet  Google Scholar 

  41. Tortrat, A.: Structure des lois indéfiniment divisibles \((\mu \in {\cal{I}}={\cal{I}}(X))\) dans un espace vectoriel topologique (séparé) X, 1967 Symposium on Probability Methods in Analysis (Loutraki, 1966), pp. 299–328. Springer (1967)

  42. Tortrat, A.: Sur la structure des lois indéfiniment divisibles dans les espaces vectoriels. Z. Wahrsch. Verw. Gebiete 11, 311–326 (1969)

    Article  Google Scholar 

  43. Tortrat, A.: Lois \(e(\lambda )\) dans les espaces vectoriels et lois stables. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 37(2), 179–182 (1976)

    Article  MathSciNet  Google Scholar 

  44. Tortrat, A.: Sur le support des lois indéfiniment divisibles dans les espaces vectoriels localement convexes. Ann. Inst. Henri Poincaré Sect. B (N.S.) 13(1), 27–43 (1977)

    MATH  Google Scholar 

  45. Trèves, F.: Topological Vector Spaces, Distributions and Kernels. Pure and Applied Mathematics. Academic Press, New York (1967)

    MATH  Google Scholar 

  46. Üstünel, A.S.: Additive processes on nuclear spaces. Ann. Probab. 12(3), 858–868 (1984)

    Article  MathSciNet  Google Scholar 

  47. Vakhania, N.N., Tarieladze, V.I., Chobanyan, S.A.: Probability Distributions on Banach Spaces. Reidel Publishing, Dordrecht (1987)

    Book  Google Scholar 

  48. Varadhan, S.R.S.: Limit theorems for sums of independent random variables with values in a Hilbert space. Sankhyā Ser. A 24, 213–238 (1962)

    MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The author would like to thank David Applebaum for all his helpful comments and suggestions. Thanks also to the University of Costa Rica for providing financial support through the Grant 820-B6-202 “Ecuaciones diferenciales en derivadas parciales en espacios de dimensión infinita.” Some earlier parts of this work were carried out at the University of Sheffield and the author wishes to express his gratitude. Many thanks are also due to the referees, who made very helpful remarks.

Author information

Authors and Affiliations

  1. Escuela de Matemática, Universidad de Costa Rica, San José, 11501-2060, Costa Rica

    C. A. Fonseca-Mora

Authors
  1. C. A. Fonseca-Mora
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to C. A. Fonseca-Mora.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Fonseca-Mora, C.A. Lévy Processes and Infinitely Divisible Measures in the Dual of a Nuclear Space. J Theor Probab 33, 649–691 (2020). https://doi.org/10.1007/s10959-019-00972-3

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10959-019-00972-3

Keywords

Mathematics Subject Classification (2010)

Search

Navigation