Given a one dimensional Lévy process $X_t$ with characteristic exponent $\psi(\xi)$, so that \begin{align} \mathbf{E}[e^{i \xi X_t}]= e^{t \psi(\xi)} \end{align} for example we can find $\psi(\xi)$ from Lévy–Khintchine representation.
What can I say on the function $1_{X_t \in A}$? Its Laplace transform? Where $1_{X_t \in A}$ is the indicator function, then it is $1$ if $X_t \in A$, $0$ otherwise. $A$ can be an interval or a point of $\mathbb{R}$. From a probabilistic point of view, the knowledge of $\psi(\xi)$ allows us to understand which is the sojourn time (or local time) of the process $X_t$ in $A$ ?
Thank you very much!