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

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

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 \widehat{\mathop{\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)