Transformation Groups on White Noise Functionals and Their Applications

In this paper we first construct a two-parameter transformation group G on the space of test white noise functionals in which the adjoints of Kuo's Fourier and Kuo's Fourier—Mehler transforms are included. Next we show that the group G is a two-dimensional complex Lie group whose infinitesimal generators are the Gross Laplacian ΔG and the number operator N , and then find an explicit description of a differentiable one-parameter subgroup of G whose infinitesimal generator is aΔG +bN . As an application, we study the solution and fundamental solution for the Cauchy problem associated with aΔ G +bN . Finally we show that each element of the adjoint group G* of G can be characterized in terms of differentiation and multiplication operators.