Université Paris 6
Pierre et Marie Curie | Université Paris 7
Denis Diderot | |

CNRS U.M.R. 7599
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``Probabilités et Modèles Aléatoires''
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**Auteur(s): **

**Code(s) de Classification MSC:**

- 60G99 None of the above but in this section
- 60H05 Stochastic integrals
- 60J65 Brownian motion, See also {58G32}

**Résumé:** For $n\geq 1$, consider a standard Brownian sheet $X$\ on $\left[ 0,1\right]
^{n}$: given a cube $R\subset \left[ 0,1\right] ^{n}$, we show that there is
a \textit{Brownian sheet bridge} $X^{0}$ (i.e. a Gaussian process indexed by
$\left[ 0,1\right] ^{n}$ and conditioned to equal a deterministic function
on the boundary of $R$) which is naturally attached to $X$, and that
multiple stochastic integrals with respect to $X^{0}$ are not only well
defined, but also able to span the space, say $L^{2}\left( X\right) $, of
square-integrable functionals of $X$. This construction yields notably a
unitary isomorphism between $L^{2}\left( X\right) $, and the symmetric Fock
space over the subset of $L^{2}\left( \left[ 0,1\right] ^{n},du_{1}...du_{n}%
\right) $ composed of functions whose integral is zero on every cube having
at least one side in common with $R$. We realize such a program by defining
a class of bounded (Hardy's type) operators from $L^{2}\left( \left[ 0,1%
\right] ^{n},du_{1},...,du_{n}\right) $ to itself, and we show that such
operators may be used to obtain the explicit form of the time space chaotic
decomposition of any sufficiently regular functional of a standard, real
valued Brownian motion: in this way, we complete the main result of \cite{io}
and we obtain a ``time-space'' counterpart to Stroock's formulae (see \cite%
{Stroock}) for Wiener chaos.

**Mots Clés:** *Wiener Chaos ; Time-Space Chaos ; Stroock's Formulae ; Brownian Bridge ; Hardy Transform*

**Date:** 2001-04-03

**Prépublication numéro:** *PMA-648*