On conditioning a self-similar growth-fragmentation by its intrinsic area

Abstract

The genealogical structure of self-similar growth-fragmentations can be described in terms of a branching random walk. The so-called intrinsic area A arises in this setting as the terminal value of a remarkable additive martingale. Motivated by connections with some models of random planar geometry, the purpose of this work is to investigate the effect of conditioning a self-similar growth-fragmentation on its intrinsic area. The distribution of A is a fixed point of a useful smoothing transform which enables us to establish the existence of a regular density a and to determine the asymptotic behavior of a(r) as r→∞ (this can be seen as a local version of Kesten–Grincevičius–Goldie theorem’s for random affine fixed point equations in a particular setting). In turn, this yields a family of martingales from which the formal conditioning on A=r can be realized by probability tilting. We point at a limit theorem for the conditional distribution given A=r as r→∞, and also observe that such conditioning still makes sense under the so-called canonical measure for which the growth-fragmentation starts from 0.