# Positive recurrence in continuous markov process but jump chain not positively recurrent or vice-versa

I just saw a theorem that sais that a continuous markov chain $$(X_t)$$ is recurrent if and only if it's jump chain is recurrent. I was wondering if it is the same thing for the concept of positive recurrence (finite expected return time). I would say intuitively that it is not the case but what example of chain can we find where :

• The continous chain is positively recurrent but the jump chain not
• The jump chain is positively recurrent but the continuous process not

?