Université Paris 6Pierre et Marie Curie Université Paris 7Denis Diderot CNRS U.M.R. 7599 Probabilités et Modèles Aléatoires''

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

Auteur(s):

Code(s) de Classification MSC:

• 60G99 None of the above but in this section
• 60H05 Stochastic integrals
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.