A white noise approach to infinitely divisible distributions on Gel'fand triple

In this note, we apply white noise analysis to infinitely divisible distributions on a real Gel'fand triple E⊂H⊂E^∗. We first introduce an index, called Hida index, for a measure on E⊂H⊂E^∗. And then, 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 Lévy measure. Finally we prove that the Hida index of the standard compound Poisson distribution on E⊂H⊂E^∗ is exactly 1.