Abstract Wiener Space Approach to Hida Calculus (Brownian Motion).

Abstract

Let $\cal S$ be the Schwartz space of rapidly decreasing real functions on ${\rm I\!R.}$ The dual space ${\cal S}\sp\ast$ of $\cal S$ consists of tempered distributions. The inclusion maps ${\cal S}\subset{\rm L\sp2(I\!R)}\subset{\cal S}\sp\ast$ are continuous. Hida's theory of Brownian and generalized Brownian functionals is the study of functionals defined on ${\cal S}\sp\ast.$ In this dissertation, the triple ${\cal S}\subset{\rm L\sp2(I\!R)}\subset{\cal S}\sp\ast$ is replaced by an abstract Wiener space ${\rm B}\sp\ast$ $\subset$ H $\subset$ B and an abstract version of Hida's theory is developed. The Gaussian measure on ${\cal S}\sp\ast$ in Hida's calculus is replaced by the standard Gaussian measure $\mu$ on the space B. The ${\cal S}\sp\ast$ valued curve $\{\delta\sb{\rm t}; {\rm t \in I\!R}\}$ in Hida calculus is replaced by a B-valued curve $\{\Theta{\rm (t); t \in I\!R}\}.$ The coordinate system, differential operator, and Laplacian operators with respect to $\{\Theta{\rm (t)}\}$ in the Abstract Wiener space setup. Similar properties and theorems as in Hida calculus are obtained.