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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 $(0,\infty)$.

user377704
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    By the first comment to this MO post it looks like you are restricted to Levy processes of the form $X_t=at+\sigma B_t$ where $a<0$ and $B$ is a standard Brownian motion. Unfortunately $0$ is regular for $(0,\infty)$ because it is for BM. Now use Girsanov's theorem to conclude this for $X,$. – Kurt G. Sep 24 '23 at 08:12
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    Could you remind us the definition of a regular point for a Levy Process? Thanks. – Gérard Letac Sep 24 '23 at 09:16
  • Sorry to be pedantic. But according to Karatzas & Shreve $0$ is a regular boundary point of $(0,+\infty)$ if $P(\tau=0)=1$ for $\tau=\inf{t>0:X_t\in\color{red}{(-\infty,0)}},.$ – Kurt G. Sep 24 '23 at 14:51
  • Interesting. I found the above definition from the book by Kyprianou. – user377704 Sep 25 '23 at 02:14

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