Nonparametric inference on Fokker-Plank and McKean-Vlasov models

We propose a kernel-based estimator of the velocity field governing the transport and diffusion of $d$-dimensional interacting particles. Assuming the initial positions are i.i.d. with law ${\mu }_0$, we establish consistency of the estimator with an explicit mean-squared error rate of order $h^2+N^{(}-2\mathrm{/}(d+2))$, where $h$ denotes the time-discretization step and N the number of particles. The analysis covers two distinct settings: the Fokker-Planck equation, where we recover the underlying potential function, and the McKean-Vlasov equation, where we deconvolve the interaction kernel driving the mean-field dynamics.