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≥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 〈F_{T}^{(d)}〉∼2[ln(T/Δt)]^{d-1}, where T is the total duration of the motion and Δt is the minimum time lapse separating points that define a facet.

Universality and time-scale invariance for the shape of planar Lévy processes

For a broad class of planar Markov processes, viz. Lévy processes satisfying certain conditions (valid, e.g., in the case of Brownian motion and Lévy flights), we establish an exact, universal formula describing the shape of the convex hull of sample paths. We show indeed that the average number of edges joining paths' points separated by a time lapse Δτ ∈ [Δτ(1),Δτ(2)] is equal to 2 ln(Δτ(2)/Δτ(1)), regardless of the specific distribution of the process's increments and regardless of its total duration T. The formula also exhibits invariance when the time scale is multiplied by any constant. Apart from its theoretical importance, our result provides insights regarding the shape of two-dimensional objects (e.g., polymer chains) modeled by the sample paths of stochastic processes generally more complex than Brownian motion. In particular, for a total time (or parameter) duration T, the average number of edges on the convex hull ("cut off" to discard edges joining points separated by a time lapse shorter than some Δτ < T) will be given by 2 ln(T/Δτ). Thus it will only grow logarithmically, rather than at some higher pace.