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 $\mathrm{A}$ 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 $\mathrm{A}$ satisfies 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 $r\to \infty$ (this can be seen as a local version of Kesten-Grincevicius-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 $\mathrm{A}=r$ can be realized by probability tilting. We point at a limit theorem for the conditional distribution given $\mathrm{A}=r$ as $r\to \infty$, and also observe that such conditioning still makes sense under the so-called canonical measure for which the growth-fragmentation starts from $0$