  We are interested in reconstructing the initial condition of a non-linear
partial differential equation (PDE), namely the Fokker-Planck equation, from
the observation of a Dyson Brownian motion at a given time $t>0$. The
Fokker-Planck equation describes the evolution of electrostatic repulsive
particle systems, and can be seen as the large particle limit of correctly
renormalized Dyson Brownian motions. The solution of the Fokker-Planck equation
can be written as the free convolution of the initial condition and the
semi-circular distribution. We propose a nonparametric estimator for the
initial condition obtained by performing the free deconvolution via the
subordination functions method. This statistical estimator is original as it
involves the resolution of a fixed point equation, and a classical
deconvolution by a Cauchy distribution. This is due to the fact that, in free
probability, the analogue of the Fourier transform is the R-transform, related
to the Cauchy transform. In past literature, there has been a focus on the
estimation of the initial conditions of linear PDEs such as the heat equation,
but to the best of our knowledge, this is the first time that the problem is
tackled for a non-linear PDE. The convergence of the estimator is proved and
the integrated mean square error is computed, providing rates of convergence
similar to the ones known for non-parametric deconvolution methods. Finally, a
simulation study illustrates the good performances of our estimator.
