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

CNRS U.M.R. 7599
``Probabilités et Modèles Aléatoires''

From Wiener to Time-Space chaos, via conditioned Gaussian measures


Code(s) de Classification MSC:

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