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]

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

Brownian motion in white noise analysis

A version of Wiener's Brownian motion ${\displaystyle B(t)}$ is obtained by the dual pairing

${\displaystyle B(t)=\langle \omega ,1\!\!1_{0,t)}\rangle ,}$

where ${\displaystyle 1\!\!1_{0,t)}}$ is the indicator function of the interval ${\displaystyle 0,t)}$. Informally

${\displaystyle B(t)=\int _{0}^{t}\omega (t)\,dt}$

and in a generalized sense

${\displaystyle \omega (t)={\frac {dB(t)}{dt}}.}$

Hilbert spaceedit

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.

Wick polynomialsedit

An orthonormal basis in this Hilbert space, generalizing that of Hermite polynomials, is given by the so-called "Wick", or "normal ordered" polynomials ${\displaystyle \left\langle {:\omega ^{n}:},f_{n}\right\rangle }$ with ${\displaystyle {:\omega ^{n}:}\in S'(\mathbb {R} ^{n})}$ and ${\displaystyle f_{n}\in S(\mathbb {R} ^{n})}$

with normalization

${\displaystyle \int _{S'(\mathbb {R} )}\left\langle :\omega ^{n}:,f_{n}\right\rangle ^{2}\,d\mu (\omega )=n!\int f_{n}^{2}(x_{1},\ldots ,x_{n})\,d^{n}x,}$

entailing the Itô-Segal-Wiener isomorphism of the white noise Hilbert space ${\displaystyle (L^{2})}$ with Fock space:

${\displaystyle L^{2}\left(S'(\mathbb {R} ),\mu \right)\simeq \bigoplus \limits _{n=0}^{\infty }\operatorname {Sym} L^{2}(\mathbb {R} ^{n},n!\,d^{n}x).}$

The "chaos expansion"

${\displaystyle \varphi (\omega )=\sum _{n}\left\langle :\omega ^{n}:,f_{n}\right\rangle }$

in terms of Wick polynomials correspond to the expansion in terms of multiple Wiener integrals. Brownian martingales ${\displaystyle M_{t}(\omega )}$ are characterized by kernel functions ${\displaystyle f_{n}}$ depending on $$Zdroj:https://en.wikipedia.org?pojem=White_noise_analysis
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