Fourier Transform and Heat Equation in White Noise Analysis.

In this dissertation, we study the Fourier transform and heat equation in the white noise set-up. It is known that the Fourier transform ${\cal F}$ can be defined on the space (${\cal S}$)* of generalized white noise functionals. We prove that ${\cal F}$ is the adjoint of a certain linear operator on the space (${\cal S}$) of test functionals. On the other hand, it is known that if f is a bounded Lip-1 function on an abstract Wiener space with Gaussian measures $P\sb{t},$ then $u(t)=P\sb{t}f$ solves the heat equation ${d\over dt}u(t)={1\over2}\Delta\sb{G}u(t)$ with initial condition u(0) = f where $\Delta\sb{G}$ is the Gross\u27 Laplacian. We show that if $\varphi$ is in (${\cal S}$), then $u(t)=P\sb{t}\varphi$ is also in (${\cal S}$) and satisfies ${d\over dt}u(t)={1\over2}\Delta\sb{G}u(t)$ with u(0) = f. The properties of some operators are also discussed