Julien Verges: Greedy paths in a continuous environment

Abstract

Alice participates in a rather unusual Easter egg hunt: she is the only player but only has a finite time T>0 to pick eggs. Moreover, she knows in advance the eggs' locations, which are scattered according to some point process. One can show that if she follows the optimal strategy, then the number M(T) of gathered eggs follows a law of large numbers: M(T)/T converges almost surely to a deterministic constant.

Let's now add a new rule: given a fixed 0 ≤ b <1, she must be on the point (bT, 0) at the time T. Let M_b(T) denote the number of collected eggs with this new constraint. Then here again M_b(T)/T converges to a deterministic constant M_b. Moreover, the map \(b \to M_b\) is nonincreasing and concave on [0,1[. In the special case where the eggs' locations is driven by a Poisson process (i.e., their number and locations inside two disjoint domains are independent), one can show extra properties for this map. I'll first formally define the problem then give a flavour of the law of large numbers' proof, which takes inspiration from the work of (Cox–Gandolfi–Griffin–Kesten, 1993-94) on the discrete, historical version of the model. Finally, I'll explain how the stability of Poisson processes with respect to some transformations of the plane yields information on the map \(b \to M_b\).