function $f:\mathbb{R}_+\to\mathbb{R}_+$ explicitely given in terms of the characteristics of the driving process, such that $A(\epsilon)/f(\epsilon)\to 1$ as $\epsilon\to 0+$.

It is possible to tilt the law of $\mathbf{X}$ such that the Eve cell starts from size 0. Indeed, in [@BBCK16], the authors introduced a new probability measure, under which $\mathbf{X}$ initiates from a distinguished particle that grows indefinitely from the initial size 0, and evolves differently from the others which eventually die out. Let denote $A^+$ the analog of $A$ under this new measure. In this case, Theorem \[Theorem area GF\] does not apply. However, Proposition \[Theorem stationary area from 0\] shows that, $t\mapsto e^{\omega t/\alpha}A^+(e^t)$ defines a stationary process. In particular, $t^{\omega_-/\alpha}A^+(t)$ has a limit in distribution as $t\to 0+$, but has no almost sure limit. We prove in Propositions \[Theorem upper envelope A(t)\] and \[Proposition lower envelope A(t)\] that, almost surely, $A^+(t)$ deviates from $t^{\omega_-/|\alpha|}$ by at most a power of $|\log(t)|$.

One of the motivations of the present work is that these branching processes turn out to be geometrically connected to some random surfaces. The Brownian map is a random surface homeomorphic to the two-dimensional sphere that appears as the Gromov-Hausdorff scaling limit when $n\to\infty$ of 