In this paper we discuss some recent developments in the theory of generalized functionals of Brownian motion. First we give a brief summary of the
Wiener-Ito multiple Integrals. We discuss some of their basic properties, and
related functional analysis on Wiener measure space. then we discuss the generalized functionals constructed by Hida. The generalized functionals of Hida are
based on L2-Sobolev spaces, thereby, admitting only Hs, s∈R valued kernels in
the multiple stochastic integrals. These functionals are much more general than
the classical Wiener-Ito class. The more recent development, due to the author,
introduces a much more broad class of generalized functionals which are based on
Lp-Sobolev spaces admitting kernels from the spaces 𝒲p,s, s∈R. This allows
analysis of a very broad class of nonlinear functionals of Brownian motion, which
can not be handled by either the Wiener-Ito class or the Hida class. For s≤0,
they represent generalized functionals on the Wiener measure space like Schwarz
distributions on finite dimensional spaces. In this paper we also introduce some
further generalizations, and construct a locally convex topological vector space of
generalized functionals. We also present some discussion on the applications of
these results.
