Julien Randon-Furling (Centre Borelli, ENS Paris Saclay)

Salle 523, couloir 12-13, 5è étage

First passages, survival probabilities & higher-dimensional convex hulls

First I will present joint work with B. de Bruyne and S. Redner, on first-passage resetting, where the resetting of a random walk to a fixed position is triggered by a first-passage event of the walk itself. We define an optimization problem that is controlled by first-passage resetting: a cost is incurred whenever the particle is reset and a reward is obtained while the particle stays near the reset-trigger point. We derive the condition to optimize the net gain in this system, namely, the reward minus the cost. I will also talk about an extension of first-passage resetting into a minimalist dynamical model of wealth evolution and wealth sharing among N agents.

Second, I will present joint work with G. Uribe Bravo and D. Zaporozhets. We focus on the convex hull of a single multidimensional random walk, with i.i.d. steps taken from any symmetric, continuous distribution. We investigate the persistence of vertices and faces on 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.

Third, time-permitting (hence, probably not...), I will 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.

References:

de Bruyne, B., R.-F., J., Redner, S. (2020). Phys. Rev. Letters, 125(5), 050602

de Bruyne, B., R.-F., J., Redner, S. (2021). J. Stat. Mech., 2021(10), 103405

R.-F., J., Zaporozhets, D. (2024). J. Math. Sc., 281(1)

R.-F., J., Uribe Bravo, G., Zaporozhets, D. to appear (2026)