Abstract

First we focus on the convex hull of a single multidimensional random walk, with iid steps taken from any symmetric, continuous distribution. We investigate the persistence of vertices and faces on the boundary of the hull. In particular we show that the corresponding distributions are universal, and follow three regimes closely linked to the Sparre Andersen theorem for one dimensional random walks.   Second, we turn to the convex hull of several multidimensional Gaussian random walks. Explicit formulas for the expected volume and expected number of faces are derived in terms of the Gaussian persistence probabilities. Special cases include the already known results about the convex hull of a single Gaussian random walk and the d-dimensional Gaussian polytope.   Third, independently from the previous topics, we present some simple toy models involving so-called “first-passage resetting” for Brownian motion.   Co-authors: G. Uribe Bravo (topic 1), D. Zaporozhets (topics 1 & 2), B. de Bruyne (topic 3), S. Redner (topic 3)