Coupling distances between Lévy measures and applications to noise sensitivity of SDE

In this talk we aim at quantifying the maximal distance between a given time series and a minimal model, which is given in terms of a scalar SDE driven by additive Lévy noise. For this purpose we introduce a metric on the formal space of Lévy measures, which is essentially based on a family of weighted Wasserstein distances of suitably renormalized tails of the Levy measures. Due to its optimal coupling property, we call this distance a coupling distance. In the main theorem of this work we prove an estimate between the laws in path space of two Lévy diffusions driven by two Lévy processes in terms of the coefficients of the equations and the coupling distance of their Lévy triplets. These results are applied in the sequel in order to show that the law of certain paleoclimate data is close in path space to the law of an asymmetric heavy-tailed Lévy diffusion with polynomial tails for a certain index, which is of interest in the climate sciences.