then given in Proposition \[Proposition lower envelope A(t)\].

We prove Theorem \[Theorem area GF\] in Section 4. The first subsection looks at the expectation of $A(\epsilon)$. In the second subsection, we introduce a useful martingale to control the fluctuations of $A(\epsilon)$ around its expectation and prove Theorem \[Theorem area GF\].

Section 5 is devoted to the proof of Propositions \[Theorem upper envelope A(t)\] and \[Proposition lower envelope A(t)\], that are respectively upper bound and lower bound for $A^+(\epsilon)$ as $\epsilon\to 0+$. After having established Proposition \[Theorem stationary area from 0\], we prove Proposition \[Theorem upper envelope A(t)\] in the first subsection. We then turn our attention to Proposition \[Proposition lower envelope A(t)\] in the second subsection. The arguments are different and the proof is more involved.

We conclude this paper with Section 6, in which we apply our results to stable surfaces.

Self-similar growth-fragmentations {#Section self-siilar growth-fragmentations}
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#### The law of a typical cell.

For all $x>0$, let $\mathbb{P}_x$ denote the law of a positive self-similar Markov process $X=(X(t))_{t\geq 0}$ starting from $x$ and absorbed at 0. Self-similarity refers to the property that under $\mathbb{P}_x$, the law of $(X(t))_{t\geq 0}$ is the same as that of $(xX(tx^\alpha))_{t\geq 0}$ under 