Wasserstein and total variation distance between marginals of L\'evy processes

We present upper bounds for the Wasserstein distance of order $p$ between the marginals of L\'evy processes, including Gaussian approximations for jumps of infinite activity. Using the convolution structure, we further derive upper bounds for the total variation distance between the marginals of L\'evy processes. Connections to other metrics like Zolotarev and Toscani-Fourier distances are established. The theory is illustrated by concrete examples and an application to statistical lower bounds.