A brief summary of nonlinear echoes and Landau damping

In this expository note we review some recent results on Landau damping in the nonlinear Vlasov equations, focusing specifically on the recent construction of nonlinear echo solutions by the author [arXiv:1605.06841] and the associated background. These solutions show that a straightforward extension of Mouhot and Villani's theorem on Landau damping to Sobolev spaces on $\mathbb T^n_x \times \mathbb R^n_v$ is impossible and hence emphasize the subtle dependence on regularity of phase mixing problems. This expository note is specifically aimed at mathematicians who study the analysis of PDEs, but not necessarily those who work specifically on kinetic theory. However, for the sake of brevity, this review is certainly not comprehensive.Comment: Expository note for the Proceedings of the Journees EDP 2017, based on a talk given at Journees EDP 2017 in Roscoff, France. Aimed at mathematicians who study the analysis of PDEs, but not necessarily those who work specifically on kinetic theory. 16 page

Asymptotic equivalence for density estimation and gaussian white noise: An extension

The aim of this paper is to present an extension of the well-known as-ymptotic equivalence between density estimation experiments and a Gaussian white noise model. Our extension consists in enlarging the nonparametric class of the admissible densities. More precisely, we propose a way to allow densities defined on any subinterval of R, and also some discontinuous or unbounded densities are considered (so long as the discontinuity and unboundedness patterns are somehow known a priori). The concept of equivalence that we shall adopt is in the sense of the Le Cam distance between statistical models. The results are constructive: all the asymptotic equivalences are established by constructing explicit Markov kernels.Comment: 11 pages. arXiv admin note: text overlap with arXiv:1503.0453