On the starting points of wanderings of Markov processes

Let $x_t$ be a Markov process on $E$ and $D$ be a subset of $E$. We will call a wandering any connected component of the set $\{t\colon x_t\in D\}$. Denote by $\overline x_t$ the process obtained from $x_t$ by killing at the first exit time out of $D$. It is proved that, under some conditions, with probability 1, every wandering starts at a point of the Martin boundary corresponding to $\overline x_t$ (i.e. the limit in (3) exists).