Generalized functionals of Brownian motion

Abstract

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 L 2 ‐Sobolev spaces, thereby, admitting only H s , 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 L p ‐Sobolev spaces admitting kernels from the spaces 𝒲 p , 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.