CREST：スクール（Prof. Laurent DENIS）

We present a new approach to absolute continuity of laws of Poisson functionals.
The theoretical framework is that of local Dirichlet forms and more precisely the (EID) property: Energy Image Density property. The method gives rise to a new explicit calculus that we first show on some simple examples : it consists in adding a particle and taking it　back after computing the gradient. This method permits to establish absolute continuity of Poisson functionals such as Levy areas, solutions of SDE’s driven by Poisson measure...It also permits to develop a Malliavin calculus on the Poisson space. The plan of the talks is the following:

Talk 1 :The theory of local Dirichlet Forms and the (EID) property.
Talk 2: Construction of a Dirichlet form on the Poisson space and the lent particle method.
Talk 3: Application of the lent particle to prove existence of density of Poisson functionals.
Talk 4: Application of the lent particle to prove smoothness of density of solutions of SDE’s driven by a Poissson measure.

These talks are based on several joint works with N. Bouleau.

References:
[1] Bichteler K., Gravereaux J.-B., Jacod J. Malliavin Calculus for Processes with Jumps (1987).
[2] Bouleau N. and Denis L. “Energy image density property and the lent particlemethod for Poisson measures" Jour. of Functional Analysis 257 (2009) 1144-1174.
[3] Bouleau N. and Denis L. “Application of the lent particle method to Poisson driven SDE’s", in revision in Probability Theory and Related Fields.
[4] Bouleau N. and Hirsch F. Dirichlet Forms and Analysis on Wiener Space De Gruyter (1991).
[5] Fukushima M., Oshima Y. and Takeda M. Dirichlet Forms and Symmetric Markov Processes De Gruyter (1994).