No Zero Divisor for Wick Product in $(S{)}^{\ast }$}

Abstract

In White Noise Analysis (WNA), various random quantities are analyzed as elements of $(S{)}^{\ast }$, the space of Hida distributions ([1]). Hida distributions are generalized functions of white noise, which is to be naturally viewed as the derivative of the Brownian motion. On $(S{)}^{\ast }$, the Wick product is defined in terms of the $\mathcal{S}$-transform. We have found such a remarkable property that the Wick product has no zero devisors among Hida distributions. This result is a WNA version of Titchmarsh's theorem and is expected to play fundamental roles in developing the \textquotedblleft operational calculus\textquotedblright in WNA along the line of Mikusiński's version for solving differential equations.

5 pages