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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 $\int x^2K(dx)<\infty$.


Path and moment criteria from a Lévy triplet

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Every real Lévy process has a unique triplet $(a,b,K)$ with $b\geq0$ and $\int(1\wedge x^2)K(dx)<\infty$ such that
$$
\mathbb E e^{iuX_t}=\exp\left\{t\left(iua-\frac12bu^2+\int_{\mathbb R\setminus\{0\}}(e^{iux}-1-iux\mathbf1_{\{|x|\leq1\}})K(dx)\right)\right\}.
$$

= Lévy–Khintchine theorem
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 Incoming links: Lévy–Khintchine formula