Strong Markov property

At an almost surely finite stopping time, a Markov process restarts from its stopped state with the same transition law and independently of the history conditional on that state.

Let $a={\mathbb{P}}_i(T_j<T_i)$ and $b={\mathbb{P}}_j(T_i<T_j)$. Before returning to $i$, the chain visits $j$ zero times with probability $1-a$ and, for $k\ge 1$, exactly $k$ times with probabilityWhen $b>0$, the expected number of visits is $a/b$.