I am working on a paper of Elie Aidekon : ‘Speed of the biased random walk on a Galton–Watson tree’ and have a question about one transformation in a proof: \begin{align} & 1+\frac{1}{1-\lambda}+\mathbb{E} \biggl[ \frac{\beta_n(I) \mathbf{1}_{\{ \beta(j)=0 \ \forall j\neq I\}}}{\lambda-1+\sum_{i=1}^{\nu(e)}\beta_n(i)}\biggl] \\ \tag{1} \label{1} ={} & \frac{\lambda}{1-\lambda}+\mathbb{E} \biggl[ \frac{\beta_n(I) \mathbf{1}_{\{ \beta(j)=0 \ \forall j\neq I\}}}{\lambda-1+\beta_n(I)}\biggl] \\ \tag{2} \label{2} ={} &\frac{\lambda}{1-\lambda}+\mathbb{E} [\nu q^{\nu-1}] \mathbb{E} \biggl[ \frac{\beta_{n-1}(e)}{\lambda-1+\beta_{n-1}(e)}\biggl]. \end{align}
Some additional informations about it:
It holds that $\lambda<1$,
$\beta(e)=\mathcal{P}^{e}(\tau_{e_{*}}=\infty)$ is the probability that the random walk start in the root $e$ of a tree and never hits the parent $e_*$,
$\beta_n(e)=\mathcal{P}^{e}\bigl(\tau^{(n)}<\tau_{e_*}\bigl)$ is the probability that the random walk start in the root $e$ of a tree and hit level $n$ before it hit the parent $e_*$.
I absolutely don’t know where the $\frac{\lambda}{1-\lambda}$ in the \eqref{1} equation comes from and how the denominator short up to $\lambda-1+\beta_n(I)$. Have someone a tip for me?
In the \eqref{2} equation I know that the branching property is used, but I don’t see this. I just know the branching property defined by ‘lines’ and can’t see how to take this definition for markov chains (without lines).