Say that I have a recurrent time-homogenous diffusion process X (continuous strong markov process) and two points $x,y$. If $X_t$ goes from $y$, to $x$ and then back to $y$ again we denote it as a "bump".
Define the stopping times iteratively:
$t_1$ First time a bump finishes
$t_2$ The time it takes from the first bump finished till the 2nd bump finishes
and so on...
(So $t_1+...+t_n$ is the total time for the n'th bump to finish ). Is it true that they form an i.i.d sequence? It seems obvious to me, and proving independence is trivial since $X_{t_n}=y$ is constant , but somehow the stationarity keeps eluding me.
Yes the variables $t_i$ are independent and identically distributed. The independence follows directly from the Strong Markov property. That gives that $t_i$ is independent of $t_j$ for all $j < i$. The Strong Markov property and time-homogeneity together imply that $$P[X_{s + t_i} \in A | t_i] = E[X_s \in A| X_0 = y]$$ for all $s \geq 0$. That yields that the $t_i$ are identically distributed.