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.
[1] It was initiated by Takeyuki Hida in his 1975 Carleton Mathematical Lecture Notes.
[2] The term white noise was first used for signals with a flat spectrum.
The white noise probability measure
of tempered distributions has the characteristic function[3] A version of Wiener's Brownian motion
Informally and in a generalized sense Fundamental to white noise analysis is the Hilbert space generalizing the Hilbert spaces
An orthonormal basis in this Hilbert space, generalizing that of Hermite polynomials, is given by the so-called "Wick", or "normal ordered" polynomials
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.
only a "cut-off": Suitable restrictions of the kernel function
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
by setting Likewise, using one defines the "S-transform" of white noise distributions
by It is worth noting that for generalized functions
as in ,[clarification needed] 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.
is the T-transform of a (unique) Hida distribution
is analytic in the whole complex plane and of second order exponential growth, i.e.
[3][5][6]The same is true for S-transforms, and similar characterization theorems hold for the more general Kondratiev distributions.
, partial, directional derivatives exist: where
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.
of white noise distributions as a Pettis integral generalizing the Itô integral beyond adapted integrands.
In general terms, there are two features of white noise analysis that have been prominent in applications.
[7][8][9][10][11] First, white noise is a generalized stochastic process with independent values at each time.
[12] Hence it plays the role of a generalized system of independent coordinates, in the sense that in various contexts it has been fruitful to express more general processes occurring e.g. in engineering or mathematical finance, in terms of white noise.
[13][9][10] Second, the characterization theorem given above allows various heuristic expressions to be identified as generalized functions of white noise.
This is particularly effective to attribute a well-defined mathematical meaning to so-called "functional integrals".
Feynman integrals in particular have been given rigorous meaning for large classes of quantum dynamical models.
Noncommutative extensions of the theory have grown under the name of quantum white noise, and finally, the rotational invariance of the white noise characteristic function provides a framework for representations of infinite-dimensional rotation groups.