Titre :

Statistical deconvolution of the free Fokker-Planck equation at fixed time

Auteur(s) :

Maïda, Mylène [Auteur]
Laboratoire Paul Painlevé - UMR 8524 [LPP]
Dat Nguyen, Tien [Auteur]
Laboratoire de Mathématiques d'Orsay [LMO]
Pham Ngoc, Thanh Mai [Auteur]
Laboratoire de Mathématiques d'Orsay [LMO]
Rivoirard, Vincent [Auteur]
CEntre de REcherches en MAthématiques de la DEcision [CEREMADE]
Tran, Chi [Auteur]
Laboratoire Analyse et de Mathématiques Appliquées [LAMA]

Titre de la revue :

Bernoulli

Pagination :

771-802

Éditeur :

Bernoulli Society for Mathematical Statistics and Probability

Date de publication :

2022

ISSN :

1350-7265

Mot(s)-clé(s) en anglais :

PDE with random initial condition
free deconvolution
inverse problem
kernel estimation
Fourier transform
mean integrated square error
Dyson Brownian motion

Discipline(s) HAL :

Mathématiques [math]/Probabilités [math.PR]
Mathématiques [math]/Statistiques [math.ST]
Statistiques [stat]/Théorie [stat.TH]

Résumé en anglais : [en]

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 ...
Lire la suite >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.Lire moins >

Langue :

Anglais

Vulgarisation :

Non

Projet ANR :

Centre Européen pour les Mathématiques, la Physique et leurs Interactions
Models and algorithms: from the discrete to the continuous

Collections :

• Laboratoire Paul Painlevé - UMR 8524

Source :

Harvested from HAL