Abstract

Convex hull of a set of points in two dimensions roughly describes the shape of the set. In this talk, I will discuss the statistical properties of the convex hull of several stochastic processes in two dimensions. By adapting Cauchy’s formula to random curves, we develop a formalism to compute explicitly the mean perimeter and the mean area of the convex hull of arbitrary two dimensional stochastic processes of a fixed duration. Our result makes an interesting and general connection between random geometry and extreme value statistics. I will discuss two examples in detail (i) a set of n independent planar Brownian paths (ii) planar branching Brownian motion with death. The first problem has application in estimating the home range of an animal population of size n, while the second is useful to estimate the spatial extent of the outbreak of animal epidemics. Finally I will also discuss two other recent examples of planar stochastic processes: (a) active run-and-tumble process and (b) resetting Brownian motion.