Asymptotic equivalence for pure jump L\'evy processes with unknown L\'evy density and Gaussian white noise

The aim of this paper is to establish a global asymptotic equivalence between the experiments generated by the discrete (high frequency) or continuous observation of a path of a L{\'e}vy process and a Gaussian white noise experiment observed up to a time T, with T tending to $\infty$. These approximations are given in the sense of the Le Cam distance, under some smoothness conditions on the unknown L{\'e}vy density. All the asymptotic equivalences are established by constructing explicit Markov kernels that can be used to reproduce one experiment from the other.