There is a property for continuous Markov process that  each point $y$  in its state space is hit with probability one starting from any interior point $x$. 

 This property is called the regularity of continuous Markov process. For example, $X_{t}$ is the 1-dimensional brownian motion. The state space is $(-\infty, +\infty)$.
I found this concept from  the paper: on increasing continuous Markov processes by E.CINLAR. Maybe there is another name from standard text book.

 My question is as follows. Suppose $X_{t}$ is a Levy process which is not a pure jump process. This  means $\sigma\neq 0 $ in its  generating $(\sigma, \gamma, \nu）$. Is $X_{t}$ regular?

Any references are very appreciated.