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

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 $\langle {\mathcal{F}}_T^{\left(d\right)}\rangle \sim 2{\left[ln\left(T/\mathrm{\Delta }t\right)\right]}^{d-1}$, where $T$ is the total duration of the motion and $\mathrm{\Delta }t$ is the minimum time lapse separating points that define a facet.

Figure 1

A sample realization of three-dimensional Brownian motion and the facets on the boundary of its convex hull.

Figure 2

Numerical simulations for the average number of triangular facets on the convex hulls of some three-dimensional Lévy processes (with ${10}^4$ and ${10}^3$ sample paths realizations): standard Brownian motion (pluses), and a stable process with stability index 1.5 (triangles). The solid line corresponds to the exact formula Eq. (14). The inset shows the asymptotics for large values of $T/\mathrm{\Delta }t$, from Eq. (15), with numerical simulations of three-dimensional Brownian motion.