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

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

A representation result for time-space Brownian chaos


Code(s) de Classification MSC:

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.

Mots Clés: Brownian motion ; Brownian bridge ; Time-space Brownian chaos ; Multiple time-space Wiener integrals ; Chaotic time-space representation property ; Weak Brownian motion

Date: 2000-02-02

Prépublication numéro: PMA-562