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.  2005 Elsevier Inc. All rights reserved