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

TL;DR

This paper proves that the Wick product in White Noise Analysis's Hida distributions has no zero divisors, extending classical theorems and aiding the development of operational calculus for stochastic differential equations.

Contribution

It establishes a zero divisor-free property of the Wick product in the space of Hida distributions, a novel result in White Noise Analysis.

Findings

Wick product has no zero divisors among Hida distributions.

The result is an analogue of Titchmarsh's theorem in WNA.

Supports development of operational calculus in stochastic analysis.

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 Mikusi\'{n}ski's version for solving differential equations.