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Positivity and Lower Bounds for the Density of Wiener Functionals

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Abstract

We consider a functional on the Wiener space which is smooth and not degenerated in Malliavin sense and we give a criterion for the strict positivity of the density, that we can use to state lower bounds as well. The results are based on the representation of the density in terms of the Riesz transform introduced in Malliavin and Thalmaier (2006) and on the estimates of the Riesz transform given in Bally and Caramellino (Stoch Process Their Appl 121:1332–1355, 2011).

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References

  1. Aida, S., Kusuoka, S., Stroock, D.: On the support of Wiener functionals. In: Asymptotic Problems in Probability Theory: Wiener Functionals and Asymptotics (Sanda/Kyoto, 1990). Pitman Res. Notes Math. Ser., vol. 284, pp. 3–34. Longman Sci. Tech., Harlow (1993)

  2. Bally, V.: Lower bounds for the density of locally elliptic Itô processes. Ann. Probab. 34, 2406–2440 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bally, V., Caramellino, L.: Riesz transform and integration by parts formulas for random variables. Stoch. Process. Their Appl. 121, 1332–1355 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bally, V., Caramellino, L.: Tubes estimates for diffusion processes under a local Hörmander condition of order one. Working paper (2012)

  5. Bally, V., Kohatsu-Higa, A.: Lower bounds for densities of Asian type stochastic differential equations. J. Funct. Anal. 258, 3134–3164 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  6. Bally, V., Fernández, B., Meda, A.: Estimates for the probability that Itô processes remain near a path. Stoch. Process. Their Appl. 121, 2087–2113 (2011)

    Article  MATH  Google Scholar 

  7. Bally, V., Pardoux, E.: Malliavin calculus for white noise driven parabolic SPDEs. Potential Anal. 9, 27–64 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  8. Ben Arous, G., Léandre, R.: Décroissance exponentielle du noyau de la chaleur sur la diagonale. II. Probab. Theory Relat. Fields 90, 377–402 (1991)

    Article  MATH  Google Scholar 

  9. Dalang, R., Nualart, E.: Potential theory for hyperbolic SPDEs. Ann. Probab. 32, 2099–2148 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  10. Fournier, N.: Strict positivity of the solution to a 2-dimensional spatially homogeneous Boltzmann equation without cutoff. Ann. Inst. Henri Poincaré Probab. Statist. q37, 481–502 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  11. Hirsch, F., Song, S.: Criteria of positivity for the density of the law of a Wiener functional. Bull. Sci. Math. 121, 261–273 (1997)

    MathSciNet  MATH  Google Scholar 

  12. Kohatsu-Higa, A.: Lower bounds for densities of uniformly elliptic random variables on Wiener space. Probab. Theory Relat. Fields 126, 421–457 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  13. Kusuoka, S., Stroock, D.: Applications of the Malliavin calculus. III. J. Fac. Sci., Univ. Tokyo, Sect. IA, Math. 34, 391–442 (1987)

    MathSciNet  MATH  Google Scholar 

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

  15. Léandre, R.: Positivity theorem for a general manifold. Statistics and Operations Research Transactions 29, 11–25 (2005)

    MATH  Google Scholar 

  16. Malliavin, P.: Stochastic Analysis. Springer (1997)

  17. Malliavin, P., Thalmaier, A.: Stochastic Calculus of Variations in Mathematical Finance. Springer, Berlin (2006)

    MATH  Google Scholar 

  18. Millet, A., Sanz-Solé, M.: A simple proof of the support theorem for diffusion processes. In: Séminaire de Probabilités, XXVIII. Lecture Notes in Math., vol. 1583, pp. 36–48. Springer, Berlin (1994)

    Chapter  Google Scholar 

  19. Millet, A., Sanz-Solé, M.: Points of positive density for the solution to a hyperbolic SPDE. Potential Anal. 7, 623–659 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  20. Nualart, D.: The Malliavin Calculus and Related Topics, 2nd edn. Springer (2006)

  21. Nualart, E.: On the density of systems of nonlinear spatially homogeneous SPDEs. Stochastics. (2012). doi:10.1080/17442508.2011.653567

  22. Nualart, E., Quer-Sardanyons, L.: Gaussian estimates for the density of the non-linear stochastic heat equation in any space dimension. Stoch. Process. Their Appl. 122, 418–447 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  23. Shigekawa, I.: Stochastic analysis. Translations of Mathematical Monographs. Iwanami Series in Modern Mathematics, vol. 224. American Mathematical Society, Providence (2004)

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Authors and Affiliations

  1. Laboratoire d’Analyse et de Mathématiques Appliquées, UMR 8050, Université Paris-Est Marne-la-Vallée, 5 Bld Descartes, Champs-sur-Marne, 77454, Marne-la-Vallée Cedex 2, France

    Vlad Bally

  2. Dipartimento di Matematica, Università di Roma - Tor Vergata, Via della Ricerca Scientifica 1, 00133, Roma, Italy

    Lucia Caramellino

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  1. Vlad Bally
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  2. Lucia Caramellino
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Correspondence to Lucia Caramellino.

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Bally, V., Caramellino, L. Positivity and Lower Bounds for the Density of Wiener Functionals. Potential Anal 39, 141–168 (2013). https://doi.org/10.1007/s11118-012-9324-7

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  • DOI: https://doi.org/10.1007/s11118-012-9324-7

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