Setup: Let $(W_t)_t$ be Brownian Motion, $x\in \mathbb T_2$ which is a two-dimensional torus. Let further $\partial B_{r_i}(x), i=0,\dots,K,$ be some circles around $x$ where $r_0> r_1>\dots>r_K > 0$. More specifically, $$ r_i = R\left( \frac \varepsilon R \right)^\frac i K \text{ for some } R\in(0, \frac 1 2). $$
If the BM hits $\partial B_{r_1}$ we say it has started an excursion at $1$ and track the indices $i$ from there on until BM hits $\partial B_{r_0}^c(x)$, which means that the excursion goes from $1\to0$ in total. During this excursion, BM can clearly also hit any of the other circles during the excursion from $1\to0$. We then say that there is also an excursion from, let's say, $m\to n$.
Problem: Let's look at an excursion from $1\to 0$. There is the following statement that I'd like to prove:
The number of excursion from $l\to l-1$ in one excursion from $1\to 0$ is distributed like the product of a Bernoulli and an independent geometric r.v., both with parameter $\frac 1 l$. (Here, $1 <l < K$)
Now, I have already shown that the probability in my case that BM on any circle touches the outer circle before the inner one with probability $\frac 1 2.$ From here, I don't really know what to do.