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.