Random walks on GW-trees (transformation)

Question

Let $(X_n)_{n\in\mathbb{N}_0}$ be a biased Random Walk on Galton-Watson tree with $\lambda\in(\lambda_c,m)$.

How can I obtain the following equation:

$\sum_{k=0}^{n-1}\mathbb{E}_{e_*}[|X_{k+1}|-|X_k| \ | \mathcal{S}]=\sum_{k=0}^{n-1}\mathbb{E}_{e_*}[\frac{\nu(X_k)-\lambda}{\nu(X_k)+\lambda} \ | \mathcal{S}]$

$\mathcal{S}$ is here the survival set and $\mathbb{E}$ the expectation regarding the annealed probab. $\mathbb{P}$.

I thought that the sum on the left side is a sum of -1 and 1 and don’t see the link to the right side.

Answer 1

It would be more useful if you specify exactly where you are reading this and what is $e_*$. Is the walk on a GW-tree or an augmented GW tree? The underlying reason for this identity is that if a vertex $v$ has $b$ children with weight 1 each, and one parent with weight $\lambda$, then the net drift (=expected increment) of the biased RW from $v$ toward the children is $b/(b+\lambda)-\lambda/(b+\lambda)$.