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

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The white noise probability measure   on the space   of tempered distributions has the characteristic function[3]

 

Brownian motion in white noise analysis

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A version of Wiener's Brownian motion   is obtained by the dual pairing

 

where   is the indicator function of the interval  . Informally

 

and in a generalized sense

 

Hilbert space

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Fundamental to white noise analysis is the Hilbert space

 

generalizing the Hilbert spaces   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   with   and  

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. Brownian martingales   are characterized by kernel functions   depending on   only a "cut-off":

 

Suitable restrictions of the kernel function   to be smooth and rapidly decreasing in   and   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  . Examples are the Hida triple, with

 

or the more general Kondratiev triples.[4]

T- and S-transform

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Using the white noise test functions

 

one introduces the "T-transform" of white noise distributions   by setting

 

Likewise, using

 

one defines the "S-transform" of white noise distributions   by

 

It is worth noting that for generalized functions  , with kernels   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.[3][4]

Characterization theorem

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The function   is the T-transform of a (unique) Hida distribution   iff for all   the function   is analytic in the whole complex plane and of second order exponential growth, i.e.  where   is some continuous quadratic form on  .[3][5][6]

The same is true for S-transforms, and similar characterization theorems hold for the more general Kondratiev distributions.[4]

Calculus

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For test functions  , partial, directional derivatives exist:

 

where   may be varied by any generalized function  . In particular, for the Dirac distribution   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.

Stochastic integrals

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A stochastic integral, the Hitsuda–Skorokhod integral, can be defined for suitable families   of white noise distributions as a Pettis integral

 

generalizing the Itô integral beyond adapted integrands.

Applications

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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.

References

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  1. ^ Huang, Zhi-yuan; Yan, Jia-An (2000). Introduction to Infinite-Dimensional Stochastic Analysis. Dordrecht: Springer Netherlands. ISBN 9789401141086. OCLC 851373497.
  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.
  3. ^ a b c 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.
  4. ^ a b c 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.
  5. ^ 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.
  6. ^ 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.
  7. ^ 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.
  8. ^ 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.
  9. ^ a b 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.
  10. ^ a b Hida, Takeyuki; Streit, Ludwig, eds. (2017). Let us use white noise. New Jersey: World Scientific. ISBN 9789813220935. OCLC 971020065.
  11. ^ Hida, Takeyuki, ed. (2005). Stochastic Analysis: Classical and Quantum. doi:10.1142/5962. ISBN 978-981-256-526-6.
  12. ^ 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.
  13. ^ 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.