Recently one of the authors has introduced the concept of generalized Poisson functionals and discussed the differentiation, renormalization, stochastic integrals etc. ([8], [9]), analogously to the works of T. Hida ([3], [4], [5]). Here we introduce a transformation for Poisson fnnctionals with the idea as in the case of Gaussian white noise (cf. [10], [11], [12], [13]). Then we can discuss the differentiation, renormalization, multiple Wiener integrals etc. in a way completely parallel with the Gaussian case. The only one exceptional point, which is most significant, is that the multiplications are described by for the Gaussian case, for the Poisson case,as will be stated in Section 5. Conversely, those formulae characterize the types of white noises.
Summary.With the aim of treating nonlinear systems with inputs being discrete and outputs being generalized functions, generalized Poisson functional are defined and analysed, where the ~-transforms and the renormalizations play essential roles. For Poisson functionals, the differential operators with respect to a Poisson white noise/5(0, their adjoint operators and the multiplication operators by t5(0 are defined. Since these operators involve the time parameter explicitly, they can be used to obtain information concerning the Poisson functionals at each point in time. As an example, a new method for measuring the Wiener kernels of such functionals is outlined.
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