Assume $X_{t}$ is a 1-dimensional Levy process on a probability $(\Omega, \mathcal{F}, P)$. For a fixed point $x$ in the state space and fixed $t\neq 0$, what's the value of $ P(\omega: X_{t}(\omega)=x)$?

In the case of Brownian motion, $P(\omega: X_{t}(\omega)=x)=0$.

In the case of Pure Jump Levy process, for example $X_{t}$ is a Poisson process, $P(\omega: X_{t}(\omega)=x)\neq 0$.

What confuses me is the case that $X_{t}$ is a Levy process with generating triplets $(\sigma, \gamma, \nu)$. Here $\sigma\neq 0$ is for the Brownian motion part, $\nu\neq 0$ is for the pure jump part. Is $P(\omega: X_{t}(\omega)=x)\neq 0$?