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There is a property for continuous Markov process that each point $y$ in its state space is hit with positive 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 triplet $(\sigma, \gamma, \nu）$. Is $X_{t}$ regular?

Any references are very appreciated.

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A counterexample is to let $X_t$ be Brownian motion with drift. Start at any point $x$ and suppose the drift is negative. Let $N_y$ be the event that $y$ is never hit, i.e., $N_y=\{(\forall t)\, X_t < y\}$. With probability one there will be some positive value that is not hit; see e.g. this question. So $$ \mathbb P (\cup_{y\in\mathbb N}\, N_y) = 1. $$ Therefore $$ \exists y\in\mathbb N\qquad \mathbb P(N_y)>0, $$ and such a $y$ is a counterexample to regularity.

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  • $\begingroup$ Thanks your answer. It's very helpful. Maybe the way I asked or that paper is wrong. It's worthwhile to checking it carefully. $\endgroup$
    – Hengyu Zhou
    Commented Feb 22, 2014 at 9:18
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    $\begingroup$ I do not know where in the paper you found the statement "hit with probability one". I can only find "hit with positive probability". $\endgroup$
    – Stephan Sturm
    Commented Feb 22, 2014 at 16:39
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    $\begingroup$ (I answered the probability-one version of the question, version number 1) $\endgroup$
    – Bjørn Kjos-Hanssen
    Commented Feb 25, 2014 at 1:24

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