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.



Random walks on GW-trees (transformation)
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 $b/(b+\lambda)-\lambda/(b+\lambda)$.


