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.

arXiv commentsubmitted

References in corpus (10)

• Optimal search strategies for hidden targets
• Exact Maximal Height Distribution of Fluctuating Interfaces
• Levy--Brownian motion on finite intervals: Mean first passage time analysis
• Convex Hull of N Planar Brownian Motions: Exact Results and an Application to Ecology
• Volumetric Properties of the Convex Hull of an n-dimensional Brownian Motion
• The convex hull for a random acceleration process in two dimensions
• Convex hull of a Brownian motion in confinement
• Mean perimeter of the convex hull of a random walk in a semi-infinite medium
• Convex hull of n planar Brownian paths: an exact formula for the average number of edges
• Universality and time-scale invariance for the shape of planar Lévy processes

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• Mean perimeter and mean area of the convex hull over planar random walks
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