Infinite dimensional analysis of pure jump Lévy processes on the Poisson space

Abstract

We develop a white noise calculus for pure jump Lévy processes on the Poisson space. This theory covers the treatment of Lévy processes of unbounded variation. The starting point of the theory is the construction of a distribution space. This space has many of the same nice properties as the classical Schwartz space, but is modified in a certain way in order to be more suitable for pure jump Lévy processes. We apply Minlos's theorem to this space and obtain a white noise measure which satisfies the first condition of analyticity, and which is non-degenerate. Furthermore, we obtain generalized Charlier polynomials for all Lévy measures. We introduce Kondratiev test function and distribution spaces, the $\mathcal{S}$-transform and the Wick product. We proceed by using a transfer principle on Poisson spaces to establish a differential calculus.