The infinite dimensional analysis as a branch of mathematical sciences
was formed in the late 19th and early 20th centuries. Motivated by
problems in mathematical physics, the first steps in this field were
taken by V. Volterra, R. GateallX, P. Levy and M. Frechet, among others
(see the preface to Levy[2]). Nevertheless, the most fruitful
direction in this field is the infinite dimensional integration theory
initiated by N. Wiener and A. N. Kolmogorov which is closely related to
the developments of the theory of stochastic processes. It was Wiener
who constructed for the first time in 1923 a probability measure on the
space of all continuous functions (i. e. the Wiener measure) which
provided an ideal math- ematical model for Brownian motion. Then some
important properties of Wiener integrals, especially the
quasi-invariance of Gaussian measures, were discovered by R. Cameron and
W. Martin[l, 2, 3]. In 1931, Kolmogorov[l] deduced a second partial
differential equation for transition probabilities of Markov processes
order with continuous trajectories (i. e. diffusion processes) and thus
revealed the deep connection between theories of differential equations
and stochastic processes. The stochastic analysis created by K. Ito
(also independently by Gihman [1]) in the forties is essentially an
infinitesimal analysis for trajectories of stochastic processes. By
virtue of Ito's stochastic differential equations one can construct
diffusion processes via direct probabilistic methods and treat them as
function- als of Brownian paths (i. e. the Wiener functionals).