Marc Hoffmann: Some questions in mathematical statistics linked to evolution PDEs

Speaker
Marc Hoffmann, Univ. Paris Dauphine-PSL

Abstract
Evolution models in applications are often analyzed via PDEs, interpreted as a macroscopic description of the phenomenon of interest. This classical approach is nevertheless challenged by empirical data for model validation, especially when the phenomenon of interest does not depend on well established physical laws. In particular the statistical inference of parameters (estimation and testing) requires an underlying stochastic model. This is usually treated via some addition of noise, sometimes artificially and always a bit arbitrarily. Relying on some specific examples that appear in population biology (cell growth, human demography) or agent-based models in economy, we propose an alternative approach. Starting from a microscopic stochastic model that can be partially observed and for which the PDE of interest is a mean-field limit, the intrinsic statistical noise becomes the fluctuation between the empirical measure of the particle system and the solution of the PDE. We will outline a rigorous statistical program in this setting and will give some results on McKean-Vlasov models that emphasize the interest of our approach.