Anticipative Markovian transformations on the Poisson space

Abstract

We study in this paper anticipative transformations on the Poisson space in the framework introduced by Picard (Ann. Inst. Henri Poincare 32 (4) (1996a) 509). Those are stochastic transformations that add particles to an initial condition or remove particles to it; they may be seen as a perturbation of the initial state with respect to the finite difference gradient D introduced by Nualart-Vives (Seminaire de Probabilite XXIV, Lecture Notes in Mathematics, Vol. 1426, Springer, Berlin, 1990). We study here an analogue of the anticipative flows on the Wiener space, which is in our context a Markov process taking its values in the Poisson space Ω and look for some criterion ensuring that the image of the Poisson probability P under the transformation is absolutely continuous with respect to P . We obtain results which are close to the results of Enchev-Stroock (J. Funct. Anal. 116 (1996) 449) founded in the Wiener space case.