Let $\Gamma_0,\Gamma_1,...$ be regeneration epochs.

If $(X_n)_{n \in \mathbb{N}}$ is a $\lambda$ biased random walk on a Galton-Watson tree, than the regeneration epochs are defined as:

$\Gamma_0:=\inf\{\iota \ | X_i\neq X_{\iota} \ \forall i\leq \iota \ \text{and} \ X_{j}\neq (X_{\iota})_* \ \forall j\geq \iota \}$,

$\Gamma_k:=\inf\{\iota \ | \iota>\Gamma_{k-1}: X_i\neq X_{\iota} \ \forall i\leq \iota \ \text{and} \ X_{j}\neq (X_{\iota})_* \ \forall j\geq \iota \}$

I want to show that

$\{\Gamma_0 < \infty \}=\mathcal{S}$, where $\mathcal{S}$ is the survival set.

The statement makes sense, of course. But i do not know how to start the proof.

Maybe someone has experience in the subject area and could help me.

Greetings : Fynn