For a Brownian motion $B_t$, the evolution of the moments with $t$ obeys the simple rule: $$\mathbb{E}[|B_t|^p] = \kappa_p |t|^{p/2},$$ with $\kappa_p<\infty$. The proof only requires to remark that the random variables $\frac{B_t}{\sqrt{t}}$ are Gaussian with variance 1.

I am interested to know if, more generally, the question was studied for Lévy processes. Of course, in the general case, the moments can be infinite. When they are not, do we have an estimation of the evolution of $\mathbb{E}[|X_t|^p]$ for a Lévy process $X_t$, or at least some bounds?

Thank you for your attention.