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.

White noise measure

The white noise probability measure ${\displaystyle \mu }$ on the space ${\displaystyle S'(\mathbb {R} )}$ of tempered distributions has the characteristic function^[3]

${\displaystyle C(f)=\int _{S'(\mathbb {R} )}\exp \left(i\left\langle \omega ,f\right\rangle \right)\,d\mu (\omega )=\exp \left(-{\frac {1}{2}}\int _{\mathbb {R} }f^{2}(t)\,dt\right),\quad f\in S(\mathbb {R} ).}$

Hilbert space

Fundamental to white noise analysis is the Hilbert space

${\displaystyle (L^{2}):=L^{2}\left(S'(\mathbb {R} ),\mu \right),}$

generalizing the Hilbert spaces ${\displaystyle L^{2}(\mathbb {R} ^{n},e^{-{\frac {1}{2}}x^{2}}d^{n}x)}$ to infinite dimension.

T- and S-transform

Using the white noise test functions

${\displaystyle \varphi _{f}(\omega ):=\exp \left(i\left\langle \omega ,f\right\rangle \right)\in (S),\quad f\in S(\mathbb {R} )}$

one introduces the "T-transform" of white noise distributions ${\displaystyle \Psi }$ by setting

${\displaystyle T\Psi (f):=\left\langle \!\left\langle \Psi ,\varphi _{f}\right\rangle \!\right\rangle .}$

Likewise, using

${\displaystyle \phi _{f}(\omega ):=\exp \left(-{\frac {1}{2}}\int f^{2}(t)\,dt\right)\exp \left(-\left\langle \omega ,f\right\rangle \right)\in (S)}$

one defines the "S-transform" of white noise distributions ${\displaystyle \Psi }$ by

${\displaystyle S\Psi (f):=\left\langle \!\left\langle \Psi ,\phi _{f}\right\rangle \!\right\rangle ,\quad f\in S(\mathbb {R} ).}$

It is worth noting that for generalized functions ${\displaystyle \Psi }$, with kernels ${\displaystyle \psi _{n}}$ as in , the S-transform is just

${\displaystyle S\Psi (f)=\sum n!\left\langle \psi _{n},f^{\otimes n}\right\rangle .}$

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.^[3][4]

Calculus

For test functions ${\displaystyle \varphi \in (S)}$, partial, directional derivatives exist:

${\displaystyle \partial _{\eta }\varphi (\omega ):=\lim _{\varepsilon \rightarrow 0}{\frac {\varphi (\omega +\varepsilon \eta )-F(\omega )}{\varepsilon }}}$

where ${\displaystyle \omega }$ may be varied by any generalized function ${\displaystyle \eta }$. In particular, for the Dirac distribution ${\displaystyle \eta =\delta _{t}}$ one defines the "Hida derivative", denoting

${\displaystyle \partial _{t}\varphi (\omega ):=\lim _{\varepsilon \rightarrow 0}{\frac {\varphi (\omega +\varepsilon \delta _{t})-F(\omega )}{\varepsilon }}.}$

Gaussian integration by parts yields the dual operator on distribution space

$${\displaystyle \partial _{t}^{\ast }=-\partial _{t}+\omega (t)}$$

An infinite-dimensional gradient

${\displaystyle \nabla$ :(S)\rightarrow L^{2}(R,dt)\otimes (S)}

is given by

${\displaystyle \nabla F(t,\omega )=\partial _{t}F(\omega ).}$

The Laplacian ${\displaystyle \triangle }$ ("Laplace–Beltrami operator") with

${\displaystyle -\triangle =\int dt\;\partial _{t}^{\ast }\partial _{t}\geq 0}$

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 ${\displaystyle \Psi (t)}$ of white noise distributions as a Pettis integral

${\displaystyle \int \partial _{t}^{\ast }\Psi (t)\,dt\in (S)^{\ast },}$

generalizing the Itô integral beyond adapted integrands.

Applications

In general terms, there are two features of white noise analysis which 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.

References