Energy image density property and the lent particle method for Poisson measures

We introduce a new approach to absolute continuity of laws of Poisson functionals. It is based on the {\it energy image density} property for Dirichlet forms and on what we call {\it the lent particle method} which consists in adding a particle and taking it back after some calculation.

[1]  F. Hirsch,et al.  Formes de Dirichlet gnrales et densit des variables alatoires relles sur l'espace de Wiener , 1986 .

[2]  N. Bouleau.  Error calculus and regularity of Poisson functionals: the lent particle method , 2008, 0809.0382.

[3]  Kiyosi Itô,et al.  SPECTRAL TYPE OF THE SHIFT TRANSFORMATION OF DIFFERENTIAL PROCESSES WITH STATIONARY INCREMENTS( , 1956 .

[4]  A criterion of density for solutions of Poisson-driven SDEs , 2000 .

[5]  M. Röckner,et al.  Dirichlet forms on topological vector spaces: closability and a Cameron-Martin formula , 1990 .

[6]  D. Applebaum.  Universal Malliavin Calculus in Fock and L\'{e}vy-It\^{o} Spaces , 2008, 0808.2593.

[7]  Construction of diffusions on configuration spaces , 2000 .

[8]  Nicolas Bouleau.  Error Calculus for Finance and Physics: The Language of Dirichlet Forms , 2003 .

[9]  Admissible vectors and their associated Dirichlet forms , 1992 .

[10]  T. O’Neil.  Geometric Measure Theory , 2002 .

[11]  A. Martin-Löf.  Limit theorems for the motion of a Poisson system of independent Markovian particles with high density , 1976 .

[12]  Elements de probabilites quantiques , 1989 .

[13]  J. Picard.  Brownian excursions, stochastic integrals, and representation of Wiener functionals , 2006 .

[14]  Nicolas Bouleau,et al.  Dirichlet Forms and Analysis on Wiener Space , 1991 .

[15]  Jean Picard,et al.  On the existence of smooth densities for jump processes , 1996 .

[16]  Nicolas Privault.  A pointwise equivalence of gradients on configuration spaces , 1998 .

[17]  A. Coquio.  Forme de Dirichlet sur l'espace canonique de Poisson et applications aux équations différentielles stochastiques , 1993 .

[18]  M. Röckner,et al.  Differential geometry of Poisson spaces , 1996 .

[19]  Zhi-Ming Ma,et al.  Introduction to the theory of (non-symmetric) Dirichlet forms , 1992 .

[20]  Liming Wu.  Construction de l'operateur de malliavin sur l'espace de poisson , 1987 .

[21]  Yasushi Ishikawa,et al.  Malliavin calculus on the Wiener–Poisson space and its application to canonical SDE with jumps , 2006 .

[22]  Sergio Albeverio,et al.  Analysis and Geometry on Configuration Spaces , 1998 .

[23]  M. Rockner,et al.  The semigroup of the Glauber dynamics of a continuous system of free particles , 2004, math/0407359.

[24]  Peter Kuster,et al.  Malliavin calculus for processes with jumps , 1991 .

[25]  M. Fukushima,et al.  Dirichlet forms and symmetric Markov processes , 1994 .

[26]  Nicolas Bouleau.  Decomposition de l'energie par niveau de potentiel , 1984 .

[27]  G. Rota,et al.  STOCHASTIC INTEGRALS: A COMBINATORIAL APPROACH , 1997 .

[28]  D. Nualart,et al.  Anticipative calculus for the Poisson process based on the Fock space , 1990 .