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Anticipative Markovian transformations on the Poisson space. (English) Zbl 1059.60071
Summary: We study anticipative transformations on the Poisson space in the framework introduced by J. Picard [Ann. Inst. Henri Poincaré, Probab. Stat. 32, 509–548 (1996; Zbl 0859.60045)]. 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 D. Nualart and J. Vives [in: Séminaire de probabilités XXIV 1988/89. Lect. Notes Math. 1426, 154–165 (1990; Zbl 0701.60048)]. 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 $$\Omega$$ and look for some criterion ensuring that the image of the Poisson probability $$\mathbb P$$ under the transformation is absolutely continuous with respect to $$\mathbb P$$. We obtain results which are close to the results of O. Enchev and D. W. Stroock [J. Funct. Anal. 116, 449–477 (1996; Zbl 0804.60072)] founded in the Wiener space case.

##### MSC:
 60H07 Stochastic calculus of variations and the Malliavin calculus 60G55 Point processes (e.g., Poisson, Cox, Hawkes processes)
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##### References:
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