For stationary, homogeneous Markov processes (viz., Lévy processes, including Brownian motion) in dimension d ≥ 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 2 [ln (T /∆t)]^d−1, where T is the total duration of the motion and ∆t is the minimum time lapse separating points that define a facet.

Reference : https://journals.aps.org/pre/abstract/10.1103/PhysRevE.95.032129