# Random walks on GW-trees (transformation)

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.

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 $$b/(b+\lambda)-\lambda/(b+\lambda)$$.