Forme de Dirichlet sur un espace de Poisson. (Dirichlet form on a Poisson space) (Q1841535)

This paper deals with the initial enlargment of a filtration \({\mathcal F}_t\) with a random variable \(L\). From the work by \textit{J. Jacod} [in: Grossissements de filtrations: exemples et applications. Lect. Notes Math. 1118, 15-35 (1985; Zbl 0568.60049)], we know that things are more tractable if the \({\mathcal F}_t\)-conditional distribution of \(L\) is absolutely continuous with respect to the Lebesgue measure. The stochastic basis is here generated by a marked point process. Stochastic gradient, Malliavin operator, carré du champ operator and Dirichlet structure are defined on this basis. A sufficient condition expressed in terms of the Dirichlet structure entails the above property of the conditional distribution of \(L\).