White noise analysis

The white noise probability measure $\backslash mu$ on the space $S\text{'}(\backslash mathbb\{R\})$ of tempered distributions has the characteristic function{{Cite book|last1=Hida|first1=Takeyuki|last2=Kuo|first2=Hui-Hsiung|last3=Potthoff|first3=Jürgen|last4=Streit|first4=Ludwig|title=White Noise |language=en-gb|doi=10.1007/978-94-017-3680-0|year = 1993|isbn = 978-90-481-4260-6}}

: $C(f)=\backslash int\_\{S\text{'}(\backslash mathbb\{R\})\}\backslash exp\; \backslash left(\; i\backslash left\backslash langle\; \backslash omega\; ,f\backslash right\backslash rangle$

\right) \, d\mu (\omega )=\exp \left( -\frac{1}{2}\int_{\mathbb{R}} f^2(t) \, dt\right), \quad f\in S(\mathbb{R}).

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

: $B(t)\; =\; \backslash langle\; \backslash omega,\; 1\backslash !\backslash !1\_\{[0,t)\}\backslash rangle,$

where $1\backslash !\backslash !1\_\{[0,t)\}$ is the indicator function of the interval $[0,t)$

. Informally

: $B(t)=\backslash int\_0^t\; \backslash omega(t)\; \backslash ,\; dt$

and in a generalized sense

: $\backslash omega(t)=\backslash frac\{d\; B(t)\}\{dt\}.$

Fundamental to white noise analysis is the Hilbert space

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

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

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

with normalization

: $\backslash int\_\{S\text{'}(\backslash mathbb\{R\})\}\backslash left\backslash langle\; :\backslash omega^n:,f\_n\; \backslash right\backslash rangle^2\; \backslash ,\; d\backslash mu(\backslash omega)\; =\; n!\backslash int\; f\_\{n\}^2(x\_1,\backslash ldots,x\_n)\; \backslash ,\; d^n\; x,$

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

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

The "chaos expansion"

: $\backslash varphi(\backslash omega)\; =\backslash sum\_n\; \backslash left\backslash langle\; :\backslash omega^n:,\; f\_n\backslash right\backslash rangle$

in terms of Wick polynomials correspond to the expansion in terms of multiple Wiener integrals. Brownian martingales $M\_t(\backslash omega)$ are characterized by kernel functions $f\_n$ depending on $t$ only a "cut-off":

: $f\_n(x\_1,\backslash ldots,x\_n;t)=$

\begin{cases}

f_n (x_1,\ldots,x_n) & \text{if } i x_i\leq t, \\

0 & \text{otherwise}.

\end{cases}

Suitable restrictions of the kernel function $\backslash varphi\; \_\{n\}$ to be smooth and rapidly decreasing in $x$ and $n$ give rise to spaces of white noise test functions $\backslash varphi$, and, by duality, to spaces of generalized functions $\backslash Psi$ of white noise, with

: $\backslash left\backslash langle\; \backslash !\; \backslash left\backslash langle\; \backslash Psi\; ,\backslash varphi\; \backslash right\backslash rangle\; \backslash !\backslash right\backslash rangle$

:=\sum_n n!\left\langle \psi_n,\varphi_n \right\rangle

generalizing the scalar product in $(L^2)$. Examples are the Hida triple, with

: $\backslash varphi\; \backslash in\; (S)\backslash subset\; (L^2)\backslash subset\; (S)^\backslash ast\; \backslash ni\; \backslash Psi$

or the more general Kondratiev triples.{{Cite journal|last1=Kondrat'ev|first1=Yu.G.|last2=Streit|first2=L.|title=Spaces of White Noise distributions: constructions, descriptions, applications. I|journal=Reports on Mathematical Physics|volume=33|issue=3|pages=341–366|doi=10.1016/0034-4877(93)90003-w|year=1993|bibcode=1993RpMP...33..341K }}

Using the white noise test functions

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

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

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

\!\right\rangle .

Likewise, using

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

one defines the "S-transform" of white noise distributions $\backslash Psi$ by

: $S\backslash Psi\; (f):=\backslash left\backslash langle\; \backslash !\backslash left\backslash langle\; \backslash Psi\; ,\backslash phi\_f\backslash right\backslash rangle\backslash !$

\right\rangle,\quad f \in S(\mathbb{R}).

It is worth noting that for generalized functions $\backslash Psi$, {{clarify|text=with kernels $\backslash psi\_n$ as in ,|reason=As in what?|date=May 2018}} the S-transform is just

: $S\backslash Psi\; (f)=\backslash sum\; n!\backslash left\backslash langle\; \backslash psi\_n,f^\{\backslash otimes\; n\}\backslash right\backslash rangle.$

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.

The function $G(f)$ is the T-transform of a (unique) Hida distribution $\backslash Psi$ iff for all $f\_1,f\_2\backslash in\; S(R),$ the function $z\backslash mapsto\; G(zf\_1+f\_2)$ is analytic in the whole complex plane and of second order exponential growth, i.e.

The same is true for S-transforms, and similar characterization theorems hold for the more general Kondratiev distributions.

For test functions $\backslash varphi\; \backslash in\; (S)$, partial, directional derivatives exist:

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

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

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

Gaussian integration by parts yields the dual operator on distribution space

: $$\backslash partial\_t^\backslash ast\; =-\backslash partial\_t+\backslash omega(t)$$

An infinite-dimensional gradient

: $\backslash nabla\; :(S)\backslash rightarrow\; L^2(R,dt)\; \backslash otimes\; (S)$

is given by

: $\backslash nabla\; F(t,\backslash omega)\; =\backslash partial\_t\; F(\backslash omega).$

The Laplacian $\backslash triangle$ ("Laplace–Beltrami operator") with

: $-\backslash triangle\; =\backslash int\; dt\backslash ;\backslash partial\_t^\backslash ast\; \backslash partial\_t\; \backslash geq\; 0$

plays an important role in infinite-dimensional analysis and is the image of the Fock space number operator.

A stochastic integral, the Hitsuda–Skorokhod integral, can be defined for suitable families $\backslash Psi\; (t)$ of white noise distributions as a Pettis integral

: $\backslash int\; \backslash partial\_t^\backslash ast\; \backslash Psi\; (t)\; \backslash ,\; dt\backslash in\; (S)^\backslash ast,$

generalizing the Itô integral beyond adapted integrands.

In general terms, there are two features of white noise analysis that have been prominent in applications.{{Cite book|title=White noise analysis and quantum information|editor-last=Accardi |editor-first=Luigi |isbn=9789813225459 |location=Singapore |publisher=World Scientific Publishing |oclc=1007244903|last1 = Accardi|first1 = Luigi|last2=Chen|first2=Louis Hsiao Yun|last3=Ohya|first3=Masanori|last4=Hida|first4=Takeyuki|last5=Si|first5=Si|date=June 2017}}{{Cite book|title=Methods and applications of white noise analysis in interdisciplinary sciences |last1=Bernido |first1=Christopher C. |last2=Carpio-Bernido |first2=M. Victoria |isbn=9789814569118 |publisher=World Scientific |location=New Jersey|oclc=884440293|year = 2015}}{{Cite book|title=Stochastic partial differential equations : a modeling, white noise functional approach|date=2010|publisher=Springer|first1=Helge |last1=Holden |first2=Bernt |last2=Øksendal |last3=Ubøe |first3=Jan |author4=Tusheng Zhang |isbn=978-0-387-89488-1|edition= 2nd |location=New York|oclc=663094108}}{{Cite book|title=Let us use white noise|editor1=Hida, Takeyuki |editor2-last=Streit |editor2-first= Ludwig |year=2017 |isbn=9789813220935|location=New Jersey|oclc=971020065 |publisher=World Scientific}}{{Cite book|title=Stochastic Analysis: Classical and Quantum|language=en-US|doi=10.1142/5962|year=2005|editor1-last=Hida|editor1-first=Takeyuki|isbn=978-981-256-526-6}}

First, white noise is a generalized stochastic process with independent values at each time.{{Cite book|title=Generalized functions |volume=4, Applications of harmonic analysis |last1=Gelfand |first1=Izrail Moiseevitch |first2=Naum Âkovlevič |last2=Vilenkin |first3=Amiel |last3=Feinstein|date=1964 |publisher=Academic Press |isbn=978-0-12-279504-6|location=New York|oclc=490085153}} 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.{{Cite journal|last1=Biagini|first1=Francesca|author1-link=Francesca Biagini|last2=Øksendal|first2=Bernt|last3=Sulem|first3=Agnès|author3-link=Agnès Sulem|last4=Wallner|first4=Naomi|date=2004-01-08|title=An introduction to white–noise theory and Malliavin calculus for fractional Brownian motion|url=http://rspa.royalsocietypublishing.org/content/460/2041/347|journal=Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences|language=en|volume=460|issue=2041|pages=347–372|doi=10.1098/rspa.2003.1246|bibcode=2004RSPSA.460..347B |issn=1364-5021|hdl=10852/10633|s2cid=120225816|hdl-access=free}}

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.

{{Reflist}}

Category:Stochastic calculus

Category:Generalized functions