# Random walks on GW-trees (regeneration epochs/survival set)

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

You need to assume that the bias satisfies $$\lambda where $$m$$ is the mean offspring. Then you can find the proof in Lemma 3.3 page 253 of [1].
• Can we therefore say: For a bias $\lambda<m$ holds $\mathcal{S}\subseteq \{ \Gamma_0<\infty\}$ because of infinitely many regeneration epochs and $\{ \Gamma_0<\infty\}\subseteq \mathcal{S}$ because in the case $\Gamma_0<\infty$ the tree must be infinite? Commented Oct 3, 2021 at 12:32