Smoothness of the convex hull of planar Brownian motion
Let B(t) be a two-dimensional Brownian motion and let C(t) be the closed convex hull of the range of B on the time interval [0,t]. The authors prove that, for a given \(t>0\), with probability one, C(t) has no corners and therefore its boundary \(\partial C(t)\) is a \(C^ 1\)-curve. This fact was already stated, with a somehow non-rigorous proof, by \textit{P. Lévy} in his book ``Processus stochastiques et mouvement brownien'' (1965; Zbl 0137.116), and was rigorously proved by \textit{El Bachir} [Thèse, Univ. Toulouse (1983)]. In the present paper, this result is proved anew, and the authors also obtain some (negative) information on the modulus of continuity of the slope of the tangent to \(\partial C(t)\). Related results have been recently obtained by \textit{K. Burdzy} and \textit{L. San Martin} [Stochastic Processes Appl. (to appear)].
scientific article; zbMATH DE number 4111736