Deconvolution of the Fokker-Planck equation at fixed time

We consider the problem of reconstructing the initial condition of a non-linear PDE,

namely the Fokker-Planck equation, from the observation of a Dyson Brownian motion at a given

time t > 0. 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 applying the subordination functions method for the free

deconvolution. This statistical estimator is original as it involves the resolution of a fixed point

equation, and a classical deconvolution by a Cauchy distribution. Eventually, the consistency

of statistical estimator is proved and the mean integrated squared error (MISE) is also studied.

Moreover, some numerical simulations are in progress.

This is a work in collaboration with M. Maïda, T.M. Pham Ngoc, V. Rivoirard and V.C. Tran.