Deconvolution of the Fokker-Planck equation at fixed time


Salle séminaire M3-324
Tien-Dat Nguyen
Laboratoire de Mathématiques d’Orsay
Mercredi, 19 Février, 2020 - 17:00 - 18:00
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.