Wiener chaos decomposition

Let $U$ be a dense subspace of a separable Hilbert space $H$. The triplet $U \subset H \subset U ^ {*}$ given by the injection $i : U \rightarrow H$ is obtained by identifying $H$ with its dual, taking the dual of $i$, and endowing $U ^ {*}$, the algebraic dual of $U$, with the weak topology. For any real $\lambda$, let $\lambda H$ be the Hilbert space obtained from $H$ by multiplying the norm on $H$ by $\lambda$.

The dual of the symmetric $k$- fold tensor product $S _ {k} ( U)$ is the space $\mathop{\rm Pol} _ {k} ( U)$ of all homogeneous polynomials of degree $k$ on $U$. The value of $F _ {k} \in \mathop{\rm Pol} _ {k} ( U)$ at $u \in U$ is $F _ {k} ( u ) = \langle F _ {k} , u ^ {\otimes k } \rangle _ {k! }$. Thus, for each $k$ there is a triplet

$$\tag{a1 } S _ {k} ( U) \subset \sqrt k! S _ {k} ( H) \subset \mathop{\rm Pol} _ {k} ( U) .$$

Taking the direct sum of the internal space $S _ {k} ( U)$ and the Hilbert sum of the central spaces there results a triplet

$$\tag{a2 } S( U) \subset \mathop{\rm Fock} ( H) \subset \mathop\widehat{ {\rm Pol}} ( U),$$

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

$$\tag{a3 } \mathop{\rm Fock} ( H) = \oplus \sqrt k! S _ {k} ( H) .$$

The external space is the space $\prod _ {k} \mathop{\rm Pol} _ {k} ( U)$ of all formal power series on $U$. The value $F( u )$ at $u \in U$ of such an $F \in \mathop\widehat{ {\rm Pol}} ( U)$ is defined as $\lim\limits _ \rightarrow \sum _ {k=} 1 ^ {N} F _ {k} ( u )$, if this limit exists. For example, for any $F = \sum F _ {k} \in \mathop{\rm Fock} ( H)$ one has

$$\tag{a4 } F( u ) = \langle F, e ^ {u} \rangle ,$$

where $e ^ {u} = \sum ( k!) ^ {-} 1 u ^ {\otimes k }$.

A probabilized vector space is a structure

$$\tag{a5 } ( U \dots X \supset \Omega , {\mathsf P} )$$

where $U$ and $X$ are two spaces in duality and $X = \mathop{\rm span} ( \Omega )$ is linearly generated by the subset $\Omega$ of $X$. This subset is endowed with a Polish (or Suslin) topology such that any $u \in U$ defines a Borel function $u( \omega ) = \langle u , \omega \rangle$ on $\Omega$. The space $U$ contains a countable subset separating the points of $\Omega$( so that the Borel $\sigma$- field is generated by $U$). Finally, ${\mathsf P}$ is a probability measure on this $\sigma$- field.

Assume, moreover, that the space of cylindrical polynomials $P( \Omega ) = \mathop{\rm span} ( u ( \omega ) ^ {k} : u \in U, k = 0, 1, 2 , . . . )$ is dense in $L _ {2} ( \Omega )$. Assume that the following bilinear form on $U$ is a scalar product:

$$\tag{a6 } b( u , v) = {\mathsf E} ( [ u ( \omega )- {\mathsf E} ( u ( \omega ))] [ v( \omega ) - {\mathsf E} ( v( \omega ))]) ,$$

and let $H$ be the completion of $U$. For any $k > 0$, let $\pi _ {k}$ denote the orthogonal projection of $L _ {2} ( \Omega )$ with range $\overline{ {P _ {<} k ( \Omega ) }}\;$, the closure of $\mathop{\rm span} ( u ( \omega ) ^ {j} : u \in U, j< k )$. Let $KO _ {k}$ be the orthogonal complement of $\overline{ {P _ {<} k ( \Omega ) }}\;$ in $\overline{ {P _ \leq k ( \Omega ) }}\;$. This space is called the $k$- th homogeneous chaos. The space $L _ {2} ( \Omega )$ is the Hilbert direct sum of the $KO _ {k}$. One says that $L _ {2} ( \Omega )$ admits a decomposition in chaos if for any $k$ the following mapping is isometric:

$$\sqrt k! S _ {k} ( H) \supset S _ {k} ( U) \ni \ Q \mapsto ^ { {I _ k} } Q - \pi _ {k} ( Q) \in \ KO _ {k} \subset L _ {2} ( \Omega ) .$$

The collection of these isometries for $k = 0, 1 \dots$ is an isometry $I$ whose inverse

$$\tag{a7 } L _ {2} ( \Omega ) \rightarrow ^ { {I ^ {-}} 1 } \mathop{\rm Fock} ( H) ,\ \ f \rightarrow \widehat{f} ,$$

extended to distributions on $\Omega$, is the starting point of distribution calculus on $\Omega$. Because of (a4), $\widehat{f}$ is explicitly given by

$$\tag{a8 } \widehat{f} ( u ) = \langle \widehat{f} , e ^ {u} \rangle = {\mathsf E} [ f \epsilon ^ {u} ] ,$$

where $\epsilon ^ {u} = I ^ {-} 1 ( e ^ {u} )$.

Decomposition in chaos was discovered by N. Wiener (in the case $\Omega$ 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 $I _ {k} ( f )$ 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)