A PROBABILISTIC APPROACH TO THE REPRESENTATION OF POSITIVE GENERALIZED WHITE NOISE FUNCTIONALS

Abstract

Let S′ be the space of tempered distributions on . In this paper we introduce a test functional space E_∞ consisting of real analytic functions on S′ of exponential type of order one which is shown to be equivalent to the Kondratiev space. Then, using a probabilistic approach, we give an elementary proof for the Kondratiev–Streit–Weaterkamp theorem, that is, every positive generalized white noise functional on E_∞ can be represented by a Borel measure on S′ satisfying exponential integrability.

^†Results presented here have partially been presented in the Ph.D dissertation of the second author.

Keywords:

• Positive generalized functions
• Measure

ACKNOWLEDGMENT

First author supported by the National Science Council of R.O.C. (Taiwan).

Notes

^†Results presented here have partially been presented in the Ph.D dissertation of the second author.