Several Applications Of K-S White Noise Analysis

In this paper, we first introduce the notion of an infinitely divisible distribution of type (T0) on a Gel'fand triple E (?) H (?) E*. Existence and some properties of such a distribution are obtained.Secondly, we introduce an index, called Hida index, for a measure on E (?) H (?) E*. And under some mild conditions, we obtain a general inequality which indicates a connection between the Hida index of an infinitely divisible distribution on E (?) H (?) E* and that of its Levy measure. Furthermore, under some conditions on the generating triple (z,B,a), we get a series of infinitely divisible distributions which admits the property that every distribution in it is the Levy measure of the next one.Finally, we discuss the quantum integral equation of Volterra type of B-valued generalized functional of white noise. Existence and uniqueness of solutions are established, explicit expression of the solution is given, the strong continuity, continuous dependence on free terms are proved.