I have read a few books about L'evy processes and tried to find a concrete example such that $\{X_t\}$ is a L'evy process (but not a compound Poisson) drifting to $-\infty$ and $0$ is not regular for $(0,\infty)$. So far, all I could find are a lot of conditions. Could anyone please give me a hint or any references? Thank you very much.
Definition of a regular point: Let $X_0=0$ and $\tau=\inf\{t>0:X_t\in(0,\infty)\}$. If $P(\tau=0)=1$, then $0$ is regular for $(0,\infty)$; If $P(\tau=0)=0$, then $0$ is not regular for $(0,\infty)$.