Wiener chaos decomposition

Let be a dense subspace of a separable Hilbert space . The triplet given by the injection is obtained by identifying with its dual, taking the dual of , and endowing , the algebraic dual of , with the weak topology. For any real , let be the Hilbert space obtained from by multiplying the norm on by .

The dual of the symmetric -fold tensor product is the space of all homogeneous polynomials of degree on . The value of at is . Thus, for each there is a triplet

(a1)

Taking the direct sum of the internal space and the Hilbert sum of the central spaces there results a triplet

(a2)

called dressed Fock space. The middle term is the usual Fock space

(a3)

The external space is the space of all formal power series on . The value at of such an is defined as , if this limit exists. For example, for any one has

(a4)

where .

A probabilized vector space is a structure

(a5)

where and are two spaces in duality and is linearly generated by the subset of . This subset is endowed with a Polish (or Suslin) topology such that any defines a Borel function on . The space contains a countable subset separating the points of (so that the Borel -field is generated by ). Finally, is a probability measure on this -field.

Assume, moreover, that the space of cylindrical polynomials is dense in . Assume that the following bilinear form on is a scalar product:

(a6)

and let be the completion of . For any , let denote the orthogonal projection of with range , the closure of . Let be the orthogonal complement of in . This space is called the -th homogeneous chaos. The space is the Hilbert direct sum of the . One says that admits a decomposition in chaos if for any the following mapping is isometric:

The collection of these isometries for is an isometry whose inverse

(a7)

extended to distributions on , is the starting point of distribution calculus on . Because of (a4), is explicitly given by

(a8)

where .

Decomposition in chaos was discovered by N. Wiener (in the case is Wiener space), [a1]. Further contributions are due to Th.A. Dwyer and I. Segal ([a2], [a3]) and these have been important for constructive quantum field theory. K. Itô obtained a decomposition into chaos for Poisson probability spaces and interpreted as iterated stochastic integrals. For formula (a8), extended to distributions for Gaussian probability spaces, cf. [a5], [a6], [a7], [a9], [a10]. There are links with Malliavin calculus, [a8].

For more material cf. e.g. also [a11], [a12]; Wick product and White noise analysis, and the references therein.

References

[a1]	N. Wiener, "The homogeneous chaos" Amer. J. Math. , 60 (1938) pp. 897–936
[a2]	Th.A., III Dwyer, "Partial differential equations in Fischer–Fock spaces for the Hilbert–Schmidt holomorphy type" Bull. Amer. Math. Soc. , 77 (1971) pp. 725–730
[a3]	I. Segal, "Tensor algebras over Hilbert spaces, I" Trans. Amer. Math. Soc. , 81 (1956) pp. 106–134
[a4]	K. Itô, "Multiple Wiener integral" J. Math. Soc. Japan (1951) pp. 157–169
[a5]	P. Krée, "Solutions faibles d'equations aux dérivées fonctionelles II" , Sem. P. Lelong 1973/1974 , Lect. notes in math. , 474 , Springer (1974) pp. 16–47
[a6]	P. Krée, R. Raczka, "Kernels and symbols of operators in quantum field theory" Ann. Inst. H. Poincaré (1978)
[a7]	B. Lascar, "Propriétés locales des espaces de type Sobolev en dimension infinie" Comm. Partial Diff. Eq. , 1 : 6 (1976) pp. 561–584
[a8]	D. Ocone, "Malliavin calculus and stochastic integral representation of functionals of diffusion processes" Stochastics , 12 (1984) pp. 161–185
[a9]	M. Krée, "Propriété de trace en dimension infinie d'espaces du type Sobolev" C.R Acad. Sci. Paris , 279 (1974) pp. 157–160
[a10]	M. Krée, "Propriété de trace en dimension infinie d'espaces de type Sobolev" Bull. Soc. Math. de France , 105 (1977) pp. 141–163
[a11]	G. Kallianpur, "The role of reproducing kernel Hilbert spaces in the study of Gaussian processes" P. Ney (ed.) , Advances in probability and related topics , 2 , M. Dekker (1970) pp. 49–84
[a12]	J. Neveu, "Processus aléatoires Gaussiens" , Univ. Montréal (1968)