Consider a Brownian Motion $(W_t)_t$ and to some $x\in\mathbb T_2$, where $\mathbb T_2$ is a two-dimensional torus, the circles $\partial B_{r_i}(x)$ around $x$ with radii $r_i= R(\frac \varepsilon R)^{i/K}$ for some $K\in\mathbb N$ and some $R\in(0,\frac 1 2), i=0,1,\dots, K.$
Now, let $1<l<K,l\in\mathbb N$. Then
The number of excursions from $l$ to $l-1$ in one excursion from $1$ to $0$ is distributed like the product of a Bernoulli distributed and an independent geometrically distributed random variable.
For clarification: One excursion from $1$ to $0$ means that $(W_t)_t$ comes into $ B_{r_1}(x)$, then spends some time there (and maybe any $\partial B_{r_i}(x)$ for some $i=2,\dots,K$), and then finally comes to $B_{r_0}^c(x)$ and the excursion ends.
Now I am wondering if this is correct and how to rigorously show that it is actually true.
