MAXIMAL BROWNIAN MOTIONS

Abstract : Let $Z=(X,Y)$ be a planar Brownian motion, $\ZZ$ the filtration it generates, and $B$ a linear Brownian motion in the filtration $\ZZ$. One says that $B$ (or its filtration) is maximal if no other linear $\ZZ$-Brownian motion has a filtration strictly bigger than that of $B$. For instance, it is shown in~\cite{Brossard - Leuridan} that $B$ is maximal if there exists a linear Brownian motion $C$ independent of $B$ and such that the planar Brownian motion $(B,C)$ generates the same filtration $\ZZ$ as $Z$. We give a necessary condition for $B$ to be maximal, and a sufficient condition which may be weaker than the existence of such a $C$. This sufficient condition is used to prove that the linear Brownian motion $\int(X\D Y-Y\D X)/|Z|$, which governs the angular part of $Z$, is maximal.
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Jean Brossard, Michel Emery, Christophe Leuridan. MAXIMAL BROWNIAN MOTIONS. Annales de l'Institut Henri Poincaré (B) Probabilités et Statistiques, Institute Henri Poincaré, 2009, 45 (3), pp.876-886. ⟨10.1214/08-AIHP194⟩. ⟨hal-00337008⟩

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