Asymptotic equivalence of Lévy density estimation 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\'evy 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\'evy density. All the asymptotic equivalences are established by constructing explicit Markov kernels that can be used to reproduce one experiment from the other.