Consider a standard unidimensional Brownian Motion $B_t$ (Wiener process).
Fact: This process returns to the origin infinite number of times with probability one.
Consider a stopping time $\tau = \inf \{t>0: B_t=0 \}$ (time of the first return).
Question: what can we say about the distribution of $\tau$? In particular, is it true that $\forall t >0$: $\Pr(\tau < t)=1$, i.e. BM almost surely returns to the origin infinitely soon?
Let $C_t = \dfrac{B_{at}}{\sqrt{a}}$. Then $C_t$ has the same distribution as $B_t$ and for $0<t_1<\cdots<t_n$, $(C_{t_1},\ldots,C_{t_n})$ has the same distribution as $(B_{t_1},\ldots,B_{t_n})$.
$C_t=0$ if and only if $B_{at}=0$. Thus rescaling the set of all returns to $0$ results in a set of all returns to $0$ that has the same probability distribution.