ON IRREDUCIBILITY OF THE ENERGY REPRESENTATION OF THE GAUGE GROUP AND THE WHITE NOISE DISTRIBUTION THEORY

We consider the energy representation for the gauge group. The gauge group is the set of C∞-mappings from a compact Riemannian manifold to a semi-simple compact Lie group. In this paper, we obtain irreducibility of the energy representation of the gauge group for any dimension of M. To prove irreducibility for the energy representation, we use the fact that each operator from a space of test functionals to a space of generalized functionals is realized as a series of integral kernel operators, called the Fock expansion.

WHITE NOISE ANALYSIS ON A NEW SPACE OF HIDA DISTRIBUTIONS

Let {α(n); n≥0} be a sequence of positive numbers satisfying certain conditions. A Gel'fand triple [Formula: see text] associated with the sequence {α (n); n ≥ 0} has been introduced on a white noise space (ℰ′,μ) by Cochran, Kuo and Sengupta. In this paper we obtain additional conditions on the sequence {α(n); n≥0} in order to carry out white noise distribution theory on the space (ℰ′,μ). Moreover, we show that the Bell numbers satisfy these additional conditions.

Transformations on white noise functions associated with second order differential operators of diagonal type

A generalized number operator and a generalized Gross Laplacian are introduced on the basis of white noise distribution theory. The equicontinuity is examined and associated one-parameter transformation groups are constructed. An infinite dimensional analogue of ax + b group and Cauchy problems on white noise space are discussed.