We are interested in reconstructing the initial condition of the free Fokker-Planck equation from the observation of a Dyson Brownian motion at a given time
t>0. We propose a nonparametric estimator of the density of the initial condition obtained by performing the free deconvolution thanks to the subordination method for which probabilistic tools have been developed recently by Arizmendi, Tarrago and Vargas (2020). Our statistical procedure is original as it involves the resolution of a fixed point equation and a (classical) deconvolution by a Cauchy distribution. We obtain convergence rates for the quadratic risk for Sobolev regularity classes and super-smooth densities. To this end, the key tool is the study of the fluctuation of random matrices. Eventually, we show that our procedure performs numerically nicely.
This is a joint work with M. Maïda, T. D. Nguyen, V. Rivoirard and V. C. Tran.