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White noise measure

The white noise probability measure μ {\displaystyle \mu } on the space S ′ ( R ) {\displaystyle S'(\mathbb {R} )} of tempered distributions has the characteristic function³

C ( f ) = ∫ S ′ ( R ) exp ⁡ ( i ⟨ ω , f ⟩ ) d μ ( ω ) = exp ⁡ ( − 1 2 ∫ R f 2 ( t ) d t ) , f ∈ S ( R ) . {\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 B ( t ) {\displaystyle B(t)} is obtained by the dual pairing

B ( t ) = ⟨ ω , 1 1 [ 0 , t ) ⟩ , {\displaystyle B(t)=\langle \omega ,1\!\!1_{[0,t)}\rangle ,}

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

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

and in a generalized sense

ω ( t ) = d B ( t ) d t . {\displaystyle \omega (t)={\frac {dB(t)}{dt}}.}

Hilbert space

Fundamental to white noise analysis is the Hilbert space

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

generalizing the Hilbert spaces L 2 ( R n , e − 1 2 x 2 d n x ) {\displaystyle L^{2}(\mathbb {R} ^{n},e^{-{\frac {1}{2}}x^{2}}d^{n}x)} 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 ⟨ : ω n : , f n ⟩ {\displaystyle \left\langle {:\omega ^{n}:},f_{n}\right\rangle } with : ω n : ∈ S ′ ( R n ) {\displaystyle {:\omega ^{n}:}\in S'(\mathbb {R} ^{n})} and f n ∈ S ( R n ) {\displaystyle f_{n}\in S(\mathbb {R} ^{n})}

with normalization

∫ S ′ ( R ) ⟨ : ω n : , f n ⟩ 2 d μ ( ω ) = n ! ∫ f n 2 ( x 1 , … , x n ) d n x , {\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 ( L 2 ) {\displaystyle (L^{2})} with Fock space:

L 2 ( S ′ ( R ) , μ ) ≃ ⨁ n = 0 ∞ Sym ⁡ L 2 ( R n , n ! d n x ) . {\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"

φ ( ω ) = ∑ n ⟨ : ω n : , f n ⟩ {\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 M t ( ω ) {\displaystyle M_{t}(\omega )} are characterized by kernel functions f n {\displaystyle f_{n}} depending on t {\displaystyle t} only a "cut-off":

f n ( x 1 , … , x n ; t ) = { f n ( x 1 , … , x n ) if  i x i ≤ t , 0 otherwise . {\displaystyle f_{n}(x_{1},\ldots ,x_{n};t)={\begin{cases}f_{n}(x_{1},\ldots ,x_{n})&{\text{if }}ix_{i}\leq t,\\0&{\text{otherwise}}.\end{cases}}}

Gelfand triples

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

⟨ ⟨ Ψ , φ ⟩ ⟩ := ∑ n n ! ⟨ ψ n , φ n ⟩ {\displaystyle \left\langle \!\left\langle \Psi ,\varphi \right\rangle \!\right\rangle :=\sum _{n}n!\left\langle \psi _{n},\varphi _{n}\right\rangle }

generalizing the scalar product in ( L 2 ) {\displaystyle (L^{2})} . Examples are the Hida triple, with

φ ∈ ( S ) ⊂ ( L 2 ) ⊂ ( S ) ∗ ∋ Ψ {\displaystyle \varphi \in (S)\subset (L^{2})\subset (S)^{\ast }\ni \Psi }

or the more general Kondratiev triples.⁴

T- and S-transform

Using the white noise test functions

φ f ( ω ) := exp ⁡ ( i ⟨ ω , f ⟩ ) ∈ ( S ) , f ∈ S ( R ) {\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 Ψ {\displaystyle \Psi } by setting

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

Likewise, using

ϕ f ( ω ) := exp ⁡ ( − 1 2 ∫ f 2 ( t ) d t ) exp ⁡ ( − ⟨ ω , f ⟩ ) ∈ ( S ) {\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 Ψ {\displaystyle \Psi } by

S Ψ ( f ) := ⟨ ⟨ Ψ , ϕ f ⟩ ⟩ , f ∈ S ( R ) . {\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 Ψ {\displaystyle \Psi } , with kernels ψ n {\displaystyle \psi _{n}} as in , the S-transform is just

S Ψ ( f ) = ∑ n ! ⟨ ψ n , f ⊗ n ⟩ . {\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.⁵⁶

Characterization theorem

The function G ( f ) {\displaystyle G(f)} is the T-transform of a (unique) Hida distribution Ψ {\displaystyle \Psi } iff for all f 1 , f 2 ∈ S ( R ) , {\displaystyle f_{1},f_{2}\in S(R),} the function z ↦ G ( z f 1 + f 2 ) {\displaystyle z\mapsto G(zf_{1}+f_{2})} is analytic in the whole complex plane and of second order exponential growth, i.e. | G (   f ) | < a e b K ( f , f ) , {\displaystyle \left\vert G(\ f)\right\vert <ae^{bK(f,f)},} where K {\displaystyle K} is some continuous quadratic form on S ′ ( R ) × S ′ ( R ) {\displaystyle S'(\mathbb {R} )\times S'(\mathbb {R} )} .⁷⁸⁹

The same is true for S-transforms, and similar characterization theorems hold for the more general Kondratiev distributions.¹⁰

Calculus

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

∂ η φ ( ω ) := lim ε → 0 φ ( ω + ε η ) − F ( ω ) ε {\displaystyle \partial _{\eta }\varphi (\omega ):=\lim _{\varepsilon \rightarrow 0}{\frac {\varphi (\omega +\varepsilon \eta )-F(\omega )}{\varepsilon }}}

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

∂ t φ ( ω ) := lim ε → 0 φ ( ω + ε δ t ) − F ( ω ) ε . {\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

∂ t ∗ = − ∂ t + ω ( t ) {\displaystyle \partial _{t}^{\ast }=-\partial _{t}+\omega (t)}

An infinite-dimensional gradient

∇ : ( S ) → L 2 ( R , d t ) ⊗ ( S ) {\displaystyle \nabla :(S)\rightarrow L^{2}(R,dt)\otimes (S)}

is given by

∇ F ( t , ω ) = ∂ t F ( ω ) . {\displaystyle \nabla F(t,\omega )=\partial _{t}F(\omega ).}

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

− △ = ∫ d t ∂ t ∗ ∂ t ≥ 0 {\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–Skorokhod integral, can be defined for suitable families Ψ ( t ) {\displaystyle \Psi (t)} of white noise distributions as a Pettis integral

∫ ∂ t ∗ Ψ ( t ) d t ∈ ( S ) ∗ , {\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 that have been prominent in applications.^1112131415

First, white noise is a generalized stochastic process with independent values at each time.¹⁶ 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.¹⁷¹⁸¹⁹

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

1. Huang, Zhi-yuan; Yan, Jia-An (2000). Introduction to Infinite-Dimensional Stochastic Analysis. Dordrecht: Springer Netherlands. ISBN 9789401141086. OCLC 851373497. 9789401141086 ↩

2. Hida, Takeyuki (1976). "Analysis of Brownian functionals". Stochastic Systems: Modeling, Identification and Optimization, I. Mathematical Programming Studies. Vol. 5. Springer, Berlin, Heidelberg. pp. 53–59. doi:10.1007/bfb0120763. ISBN 978-3-642-00783-5. 978-3-642-00783-5 ↩

3. Hida, Takeyuki; Kuo, Hui-Hsiung; Potthoff, Jürgen; Streit, Ludwig (1993). White Noise. doi:10.1007/978-94-017-3680-0. ISBN 978-90-481-4260-6. 978-90-481-4260-6 ↩

4. Kondrat'ev, Yu.G.; Streit, L. (1993). "Spaces of White Noise distributions: constructions, descriptions, applications. I". Reports on Mathematical Physics. 33 (3): 341–366. Bibcode:1993RpMP...33..341K. doi:10.1016/0034-4877(93)90003-w. /wiki/Bibcode_(identifier) ↩

5. Hida, Takeyuki; Kuo, Hui-Hsiung; Potthoff, Jürgen; Streit, Ludwig (1993). White Noise. doi:10.1007/978-94-017-3680-0. ISBN 978-90-481-4260-6. 978-90-481-4260-6 ↩

6. Kondrat'ev, Yu.G.; Streit, L. (1993). "Spaces of White Noise distributions: constructions, descriptions, applications. I". Reports on Mathematical Physics. 33 (3): 341–366. Bibcode:1993RpMP...33..341K. doi:10.1016/0034-4877(93)90003-w. /wiki/Bibcode_(identifier) ↩

7. Hida, Takeyuki; Kuo, Hui-Hsiung; Potthoff, Jürgen; Streit, Ludwig (1993). White Noise. doi:10.1007/978-94-017-3680-0. ISBN 978-90-481-4260-6. 978-90-481-4260-6 ↩

8. Kuo, H.-H.; Potthoff, J.; Streit, L. (1991). "A characterization of white noise test functionals". Nagoya Mathematical Journal. 121: 185–194. doi:10.1017/S0027763000003469. ISSN 0027-7630. https://projecteuclid.org/euclid.nmj/1118782788 ↩

9. Kondratiev, Yu.G.; Leukert, P.; Potthoff, J.; Streit, L.; Westerkamp, W. (1996). "Generalized Functionals in Gaussian Spaces: The Characterization Theorem Revisited". Journal of Functional Analysis. 141 (2): 301–318. arXiv:math/0303054. doi:10.1006/jfan.1996.0130. S2CID 58889052. /wiki/ArXiv_(identifier) ↩

10. Kondrat'ev, Yu.G.; Streit, L. (1993). "Spaces of White Noise distributions: constructions, descriptions, applications. I". Reports on Mathematical Physics. 33 (3): 341–366. Bibcode:1993RpMP...33..341K. doi:10.1016/0034-4877(93)90003-w. /wiki/Bibcode_(identifier) ↩

11. Accardi, Luigi; Chen, Louis Hsiao Yun; Ohya, Masanori; Hida, Takeyuki; Si, Si (June 2017). Accardi, Luigi (ed.). White noise analysis and quantum information. Singapore: World Scientific Publishing. ISBN 9789813225459. OCLC 1007244903. 9789813225459 ↩

12. Bernido, Christopher C.; Carpio-Bernido, M. Victoria (2015). Methods and applications of white noise analysis in interdisciplinary sciences. New Jersey: World Scientific. ISBN 9789814569118. OCLC 884440293. 9789814569118 ↩

13. Holden, Helge; Øksendal, Bernt; Ubøe, Jan; Tusheng Zhang (2010). Stochastic partial differential equations : a modeling, white noise functional approach (2nd ed.). New York: Springer. ISBN 978-0-387-89488-1. OCLC 663094108. 978-0-387-89488-1 ↩

14. Hida, Takeyuki; Streit, Ludwig, eds. (2017). Let us use white noise. New Jersey: World Scientific. ISBN 9789813220935. OCLC 971020065. 9789813220935 ↩

15. Hida, Takeyuki, ed. (2005). Stochastic Analysis: Classical and Quantum. doi:10.1142/5962. ISBN 978-981-256-526-6. 978-981-256-526-6 ↩

16. Gelfand, Izrail Moiseevitch; Vilenkin, Naum Âkovlevič; Feinstein, Amiel (1964). Generalized functions. Vol. 4, Applications of harmonic analysis. New York: Academic Press. ISBN 978-0-12-279504-6. OCLC 490085153. {{cite book}}: ISBN / Date incompatibility (help) 978-0-12-279504-6 ↩

17. Biagini, Francesca; Øksendal, Bernt; Sulem, Agnès; Wallner, Naomi (2004-01-08). "An introduction to white–noise theory and Malliavin calculus for fractional Brownian motion". Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences. 460 (2041): 347–372. Bibcode:2004RSPSA.460..347B. doi:10.1098/rspa.2003.1246. hdl:10852/10633. ISSN 1364-5021. S2CID 120225816. /wiki/Francesca_Biagini ↩

18. Holden, Helge; Øksendal, Bernt; Ubøe, Jan; Tusheng Zhang (2010). Stochastic partial differential equations : a modeling, white noise functional approach (2nd ed.). New York: Springer. ISBN 978-0-387-89488-1. OCLC 663094108. 978-0-387-89488-1 ↩

19. Hida, Takeyuki; Streit, Ludwig, eds. (2017). Let us use white noise. New Jersey: World Scientific. ISBN 9789813220935. OCLC 971020065. 9789813220935 ↩