Chaotic and variational calculus in discrete and continuous time for the poisson process

We study a new interpretation of the Poisson space as a triplet (H,B,P) where if is a Hilbert spaceB is the completion of H and P is the extension to the Borel σ-algebra of B of a cylindrical measure on B. A discrete chaotic decomposition of L 2(B P) is defined, along with multiple stochastic integrals of elements of H on. It turns out that the directional derivative of functionals in L 2(B P) in the direction of an element of H is an annihilation operator on the discrete chaotic decomposition. By composition with the Poisson process, we deduce continuous-time operators of derivation and divergence that form the number operator on the discrete chaotic decomposition. Those results are applied to the representation of random variables in the Wiener-Poisson chaotic decomposition