Facets on the convex hull of $d$-dimensional Brownian and Lévy motion

Abstract

For stationary, homogeneous Markov processes (viz., Lévy processes, including Brownian motion) in dimension $d\ge 3$, we establish an exact formula for the average number of $(d-1)$-dimensional facets that can be defined by $d$ points on the process's path. This formula defines a universality class in that it is independent of the increments' distribution, and it admits a closed form when $d=3$, a case which is of particular interest for applications in biophysics, chemistry and polymer science. We also show that the asymptotical average number of facets behaves as $⟨{\mathcal{F}}_T^{(d)}⟩\sim 2{\left[\ln \left(T/\Delta t\right)\right]}^{d-1}$, where $T$ is the total duration of the motion and $\Delta t$ is the minimum time lapse separating points that define a facet.

submitted

References in corpus (1)

• Convex hull of n planar Brownian paths: an exact formula for the average number of edges