White noise analysis
In probability theory, a branch of mathematics, white noise analysis, otherwise known as Hida calculus, is a framework for infinite-dimensional and stochastic calculus, based on the Gaussian white noise probability space, to be compared with Malliavin calculus based on the Wiener process. It was initiated by Takeyuki Hida in his 1975 Carleton Mathematical Lecture Notes. The term white noise was first used for signals with a flat spectrum. White noise measure The white noise probability measure on the space of tempered distributions has the characteristic function Brownian motion in white noise analysis A version of Wiener's Brownian motion is obtained by the dual pairing where is the indicator function of the interval . Informally and in a generalized sense Hilbert space Fundamental to white noise analysis is the Hilbert space generalizing the Hilbert spaces to infinite dimension. Wick polynomials An orthonormal basis in this Hilbert space, generalizing that of Hermite polynomials, is given by the so-called "Wick", or "normal ordered" polynomials with and with normalization entailing the Itô-Segal-Wiener isomorphism of the white noise Hilbert space with Fock space: The "chaos expansion" in terms of Wick polynomials correspond to the expansion in terms of multiple Wiener integrals. Brownian martingales are characterized