  In this paper, we study the compressibility of random processes and fields,
called generalized L\'evy processes, that are solutions of stochastic
differential equations driven by $d$-dimensional periodic L\'evy white noises.
Our results are based on the estimation of the Besov regularity of L\'evy white
noises and generalized L\'evy processes. We show in particular that
non-Gaussian generalized L\'evy processes are more compressible in a wavelet
basis than the corresponding Gaussian processes, in the sense that their
$n$-term approximation error decays faster. We quantify this compressibility in
terms of the Blumenthal-Getoor index of the underlying L\'evy white noise.
