Let $(X_t)_t$ be a jump Markov process with a symmetric Kernel $J:\mathbb R\times \mathbb R\to \mathbb R^+$. $J(x, y)>0$ gives the intensity of jumping from $x$ to $y$. Let $D$ be a region in $\mathbb R$ and let $\tau=\inf \{t\geq 0 :X_t\not\in D \}$ be the first time at which the process reaches the complement of $D$. I would like to prove that if the process starts from $x\in D$ then $$P^x(\tau<\infty)=1$$
I was thinking to use Borel Cantelli's theorem but I didn't succeed. Could someone help me?