Appl Math Optim 37:205–223 (1998)
1998 Springer-Verlag New York Inc.
Transformation Groups on White Noise Functionals
and Their Applications
D. M. Chung and U. C. Ji
Department of Mathematics, Sogang University,
Seoul, 121-742 Korea
Abstract. 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
and the number operator N, and then ﬁnd an explicitdescription
of a differentiable one-parameter subgroup of G whose inﬁnitesimal generator is
+ bN. As an application, we study the solution and fundamental solution for
the Cauchy problem associated with a
+bN. Finally we show that each element
of the adjoint group G
of G can be characterized in terms of differentiation and
Key Words. White noise functional, Gross Laplacian, Number operator, One-
parameter group, Inﬁnitesimal generator, Cauchy problem.
AMS Classiﬁcation. Primary 60H30, Secondary 46F25.
In recent years white noise analysis, initiated by Hida  in 1975, has been considerably
developed to an inﬁnite-dimensional analysis on Gaussian space. It has applications to
many ﬁelds: stochastic analysis, Feynman path integral, quantum physics and inﬁnite
dimensional harmonic analysis and so on, see, e.g., , , , , and the references
The research by the ﬁrst author was supported by BSRIP, MOE, Korea, 1996. This paper is a revised
version of the paper entitled “Groups of operators on white noise functionals and applications to Cauchy
problems in white noise analysis” (preprint, 1994).