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

### A representation result for time-space Brownian chaos

Auteur(s):

Code(s) de Classification MSC:

• 60G99 None of the above but in this section
• 60H05 Stochastic integrals
Résumé: Given a Brownian motion $X$, we say that a square-integrable functional $F$ belongs to the $n$-th time-space Brownian chaos if $F$ is contained in the vector space $\overline{\Pi }_{n}$, generated by r.v.'s of the form $% f_{1}\left( X_{t_{1}}\right) ...f_{n}\left( X_{t_{n}}\right)$, and $F$ is orthogonal to $\overline{\Pi }_{n-1}$. We therefore show that every element of the $n$-th Brownian chaos can be represented as a multiple time-space Wiener integral of the $n$-th order, thus proving a new representation property for Brownian motion.