Talks
Bastien MalleinA fragmentation toy-model for the perimeter of the convex hull of a Brownian motion in a disk
Consider a Brownian motion in the unit disk, reflected orthogonally at its boundary. This can be thought of as the trajectory of an animal exploring its territory, foraging for food. To estimate the domain covered by this animal during its search for t units of time, one can estimate the area, or the perimeter, of the convex hull of its positions. This question was studied by De Bruyne et al, who conjectured a convergence rate \(\Theta(t^{1/4} e^{-2 t^{1/2}})\) for these quantities, towards the area and perimeter of the unit disk respectively.
We introduce in this talk an inhomogeneous fragmentation process in which particles of mass \(m \in (0, 1)\) split at a rate proportional to \(\frac{1}{|\log m|}\), as a toy-model for the study of this speed of convergence. This process does not belong to the well-studied family of self-similar fragmentation processes. Our main results characterize the Laplace transform of the typical fragment of such a process, at any time, and its large time behavior. Based on a joint work with Bénédicte Haas.
