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Riesz transform and integration by parts formulas for random variables. (English) Zbl 1229.60069
Integration by parts formulae on the Wiener space enable to prove representation formulae for the distribution laws of variables. In particular, one can write a formula for the density of a variable \(F\), involving the expectation of \(\partial_i Q_d(F-x)\), where \(Q_d\) is the Poisson kernel on \(\mathbb R^d\setminus\{0\}\). A central point of interest in this work is the estimation of \(\mathbb E(|\partial_i Q_d(F-x)|^d)\). As a consequence, regularity and estimates for the density of \(F\) are obtained, and the set where the density is strictly positive is studied.

60H07 Stochastic calculus of variations and the Malliavin calculus
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
Full Text: DOI arXiv
[1] V. Bally, An elementary introduction to Malliavin calculus. Rapport de recherche 4718. INRIA, 2003.
[2] V. Bally, L. Caramellino, Lower bounds for the density of Ito processes under weak regularity assumptions. Preprint, 2006.
[3] Bell, D., On the relationship between differentiability and absolute continuity of measures on \(\mathbb{R}^n\), Probab. theory related fields, 72, 417-424, (1986) · Zbl 0577.28001
[4] Bouleau, N.; Hirsch, F., Dirichlet forms and analysis on the Wiener space, () · Zbl 0671.60053
[5] H. Brezis, Analyse fonctionelle. Théorie et applications. Masson, Paris, 1983.
[6] Fang, S., On the ornstein – uhlenbeck process, Stoch. stoch. rep., 46, 141-159, (1994) · Zbl 0826.60067
[7] Hirsch, F.; Song, S., Properties of the set of positivity for the density of a regular Wiener functional, Bull. sci. math., 122, 1-15, (1998) · Zbl 0897.60060
[8] Kohatsu-Higa, A.; Yasuda, K., Estimating multidimensional density functions using the malliavin – thalmaier formula, SIAM J. numer. anal., 47, 1546-1575, (2009) · Zbl 1395.62082
[9] Kohatsu-Higa, A.; Yasuda, K., Estimating multidimensional density functions for random variables in Wiener space, C.R. math. acad. sci. Paris, 346, 335-338, (2008) · Zbl 1134.60045
[10] Malliavin, P., Stochastic calculus of variations and hypoelliptic operators, (), 195-263
[11] Malliavin, P., Stochastic analysis, (1997), Springer · Zbl 0878.60001
[12] Malliavin, P.; Nualart, E., Density minoration of a strongly non degenerated random variable, J. funct. anal., 256, 4197-4214, (2009) · Zbl 1175.60055
[13] Malliavin, P.; Thalmaier, A., Stochastic calculus of variations in mathematical finance, (2006), Springer-Verlag Berlin · Zbl 1124.91035
[14] Nualart, D., The Malliavin calculus and related topics, (1995), Springer-Verlag · Zbl 0837.60050
[15] Shigekawa, I., Stochastic analysis, () · Zbl 0548.60035
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