I am reading Theorem 7.11 from the book on Levy processes by Kyprianoy. I cannot intuitively grasp this: In this theorem it is said that if the process has a Gaussian component then it creeps upwards. So this means that in the case you have a Levy process with a Guassian component and Levy measure with infinite total mass, then the processes still creeps upwards. I am wondering how this ties with the fact that jump times are dense on any time interval in this case. Doesn't the latter mean that the process won't be able to cross but only jump above any level x ? There is something that I am missing. But what is that?
A Process creeping upward means
$$\mathbb{P}(X_{\tau^+}=x)>0,\ \ x>0.$$