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

TL;DR

This paper studies how conditioning a self-similar growth-fragmentation process on its intrinsic area affects its structure, revealing asymptotic behaviors and enabling probabilistic conditioning through martingale tilting.

Contribution

It establishes the existence of a density for the intrinsic area, analyzes its asymptotics, and develops a framework for conditioning the process on this area using martingale techniques.

Findings

Density of intrinsic area exists and is regular.

Asymptotic behavior of the density as area tends to infinity.

Conditional distribution converges under large area conditioning.

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 $0$