Integral Representation of Second Quantization and Its Application to White Noise Analysis

It is shown that the second quantization Γ(K) for a continuous linear operator K on a certain nuclear space E enjoys an integral representation on the dual space E* with respect to the canonical Gaussian measure μ on E*. Employing such a representation, sharper growth estimates and locality for white noise functionals are obtained. We also establish a topological equivalence between two new spaces of test white noise functionals M and E, introduced respectively by Meyer and Yan and by Lee. It is also shown that every member in M has an analytic version in E. As a consequence of the equivalence of M and E, we show that positive generalized functionals in M can be represented by finite measures with exponentially integrable property.