Multi-parameter wiener functionals and fourier transform

Abstract

We develop the general theory of multi-parameter Wiener functional which is parallel to that of one parameter Brownian functional. We study the causal analysis of the multi-parameter Wiener functionals and show that the W(u)-multiplication can be defined on the space (L$^2$)-of generalized functionals. We introduce the Fourier transform on the space (L$^2$)-of generalized functionals of multi-parameter Wiener process. We show that the Fourier transform carries W(u)-differentiation into multiplication by iv(u) and v(u)-differentiation of Fourier transformed functional $\varphi$ is equal to the Fourier transform of -iW(u)$\varphi$.