Path and moment criteria from a Lévy triplet

For the convention in the Lévy–Khintchine formula, a Lévy process has almost surely differentiable paths exactly when $b=0$ and $K=0$, and continuous paths exactly when $K=0$. It is integrable exactly when ${\int }_{|x|>1}|x|K(dx)<\infty$, and it has a finite second moment exactly when $\int x^{2}K(dx)<\infty$.