Recall that $\Lambda$ is the Lévy measure of $\xi$. The Lévy measure of $\eta^-$, that we call $\Pi^-$, is then given by $$\begin{aligned}
\label{Equation Pi Levy measure of eta^-}
        \Pi^-(\mathrm{d}y)=e^{\omega_-y}(\Lambda+\widetilde{\Lambda})(\mathrm{d}y),
    \end{aligned}$$ where $\widetilde{\Lambda}$ is the push-forward of $\Lambda$ by the mapping $x\mapsto\log(1-e^x)\mathds{1}_{\{x<0\}}$. The Laplace exponent[^2] of $\eta^-$ is given by $$\begin{aligned}
\label{Equation Laplace exponent phi of eta^-}
        \phi_-(q)=\kappa(\omega_-+q),\quad q\in\mathbb{R}.
    \end{aligned}$$ Since $\phi_-'(0)<0$ by , we know that $Y^-$ is absorbed at 0 in finite time almost surely. By definition of Lamperti’s time-change, its absorption time $I$ is given by $$\begin{aligned}
\label{Equation definition I}
        I=\int_0^\infty\exp\left(-\alpha\eta^-(t)\right)\mathrm{d}t.
    \end{aligned}$$ Thus written, $I$ is known as an *exponential functional* of the Lévy process $\eta^-$, and this kind of random variables has been extensively studied (see e.g. [@PS16] and references therein).

One of the interests of this spinal decomposition is that it allows us to study $A$, e.g. through the forthcoming lemma \[Lemma expect A(epsilon) = distr function of I\] that states an explicit and fruitful connection between $I$ and $A$.

#### Conditioning the spine to grow indefinitely.

A second spinal decomposition was also introduced in [@BBCK16]. Under assumption , the process $n\mapsto\sum_{|u|=n}\chi_u(0)^{\omega_+}$ indexed by generation is also a martingale, but now with terminal value $0$ 