Intrinsic area of a self-similar growth-fragmentation

We study the behaviour of a natural measure de ned on the leaves of the genealogical tree of some branching processes, namely self-similar growth-fragmentation processes. Each particle, or cell, is attributed a positive mass that evolves in continuous time by randomly growing and splitting. We are interested in the mass of the ball of radius t centered at the root, denoted A(t). After giving the criterion that discerns the absolutely continuous and the singular case for t 7! A(t), we will look at the asymptotics of A(t), as t ! 0+. We will apply these results to the intrinsic volume measure of the Brownian map, exploiting the connection between growth-fragmentations and random planar maps recently obtained in [1].

This talk is based on [2] and [3].

References
[1] Jean Bertoin, Timothy Budd, Nicolas Curien, and Igor Kortchemski. Martingales in self-similar growth-fragmentations and their connections with random planar maps. Probab. Theory Related Fields, 172(3-4):663{724, 2018.
[2] Francois G. Ged. Intrinsic area near the origin for self-similar growth-fragmentations and related random surfaces. arXiv:1908.03746, 2019.
[3] Francois G. Ged. Pro le of a self-similar growth-fragmentation. Electron. J. Probab., 24:Paper No. 7, 21, 2019.