Operators of Stochastic Differentiation on Spaces of Regular Test and Generalized Functions in the Lévy White Noise Analysis

Abstract

The operators of stochastic differentiation, which are closely related with stochastic integrals and with the Hida stochastic derivative, play an important role in the classical white noise analysis. In particular, one can use these operators in order to study properties of solutions of normally ordered stochastic equations, and properties of the extended Skorohod stochastic integral. So, it is natural to introduce and to study analogs of the mentioned operators in the Levy white noise analysis. In this paper, using the theory of Hilbert equipments, in terms of the Lytvynov’s generalization of the chaotic representation property we introduce operators of stochastic differentiation on spaces from parametrized regular rigging of the space of square integrable with respect to the measure of a Levy white noise functions. Then we establish some properties of introduced operators. This gives a possibility to extend to the Levy white noise analysis and to deepen the well-known results of the classical white noise analysis that are connected with the operators of stochastic differentiation.