Abstract
In modeling, evolutionary phenomena are often analyzed through the prism of partial differential equations, interpreted as a macroscopic description of the problem of interest. However, this classic approach comes up against the problem of model validation, when experimental data are available and the phenomena studied do not follow well-identified physical laws. In particular, statistical inference of parameters (estimation, testing) requires the presence of an underlying stochastic model. This is most often dealt with by adding experimental noise - sometimes artificial, always somewhat arbitrary - intended to solve the problem.
Based on a few specific examples from population biology (human demography, cell growth) or agent models in economics, we propose a different approach. Starting with a microscopic model for which observables are available and whose PDE of interest is seen as the mean-field limit, statistical noise naturally becomes the fluctuation between the empirical measurement of the observed particle system and its limit PDE. The statistical study is more difficult, but more realistic. We will outline a non-parametric inference program within this framework and illustrate the benefits of such an approach on McKean-Vlasov-type models.
