Smoothness of the Convex Hull of Planar Brownian Motion

Cranston, M. ; Hsu, P. ; March, P.

Ann. Probab., Tome 17 (1989) no. 4, p. 144-150 / Harvested from Project Euclid

Résumé

In this article we prove that for each $t > 0$, almost surely $\partial C(t)$, the boundary of the convex hull of two dimensional Brownian motion up to time $t$, is a $C^1$ curve in the plane. We also prove that if $\eta$ is a modulus of continuity such that $x\eta(x)$ is convex and $\int^1_0\eta(x) dx/x < \infty$ then for each $t > 0$, almost surely $\partial C(t)$ is not a $C^{1, \eta}$ curve in the plane.

Détail

Publié le : 1989-01-14
Classification:  Planar Brownian motion,  convex hull,  excursion,  60G65