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 by kernel functions depending on only a "cut-off": Gelfand triples Suitable restrictions of the kernel function to be smooth and rapidly decreasing in and give rise to spaces of white noise test functions , and, by duality, to spaces of generalized functions of white noise, with generalizing the scalar product in . Examples are the Hida triple, with or the more general Kondratiev triples. T- and S-transform Using the white noise test functions one introduces the "T-transform" of white noise distributions by setting Likewise, using one defines the "S-transform" of white noise distributions by It is worth noting that for generalized functions , the S-transform is just Depending on the choice of Gelfand triple, the white noise test functions and distributions are characterized by corresponding growth and analyticity properties of their S- or T-transforms. Characterization theorem The function is the T-transform of a (unique) Hida distribution iff for all the function is analytic in the whole complex plane and of second order exponential growth, i.e. where is some continuous quadratic form on .The same is true for S-transforms, and similar characterization theorems hold for the more general Kondratiev distributions. Calculus For test functions , partial, directional derivatives exist: where may be varied by any generalized function . In particular, for the Dirac distribution one defines the "Hida derivative", denoting Gaussian integration by parts yields the dual operator on distribution space An infinite-dimensional gradient is given by The Laplacian ("Laplace–Beltrami operator") with plays an important role in infinite-dimensional analysis and is the image of the Fock space number operator. Stochastic integrals A stochastic integral, the "Hitsuda-Skorohod integral" can be defined for suitable families of white noise distributions as a Pettis integral generalizing the Itô integral beyond adapted integrands. Applications In general terms, there are two features of white noise analysis which