Let Z=(X, Y) be a planar Brownian motion, $\mathcal{Z}$ the filtration it generates, and B a linear Brownian motion in the filtration $\mathcal{Z}$ . One says that B (or its filtration) is maximal if no other linear $\mathcal{Z}$ -Brownian motion has a filtration strictly bigger than that of B. For instance, it is shown in [In Séminaire de Probabilités XLI 265–278 (2008) Springer] 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 $\mathcal{Z}$ as Z. We do not know if this sufficient condition for maximality is also necessary.
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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 ∫(X dY−Y dX)/|Z|, which governs the angular part of Z, is maximal.