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May 2022 Statistical deconvolution of the free Fokker-Planck equation at fixed time
Mylène Maïda, Tien Dat Nguyen, Thanh Mai Pham Ngoc, Vincent Rivoirard, Viet Chi Tran
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Bernoulli 28(2): 771-802 (May 2022). DOI: 10.3150/21-BEJ1366


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.


The authors thank P. Tarrago for useful discussions. M.M. acknowledges support from the Labex CEMPI (ANR-11-LABX-0007-01). T.D.N. was supported by a public grant as part of the Investissement d’avenir project, reference ANR-11-LABX-0056-LMH, LabEx LMH. V.C.T. is partly supported by Labex Bézout (ANR-10-LABX-58) and by the Chair “Modélisation Mathématique et Biodiversité” of Veolia Environnement-Ecole Polytechnique-Museum National d’Histoire Naturelle-Fondation X. The authors would like to thank the Editor and two anonymous referees for valuable comments and suggestions.


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Mylène Maïda. Tien Dat Nguyen. Thanh Mai Pham Ngoc. Vincent Rivoirard. Viet Chi Tran. "Statistical deconvolution of the free Fokker-Planck equation at fixed time." Bernoulli 28 (2) 771 - 802, May 2022.


Received: 1 June 2020; Revised: 1 January 2021; Published: May 2022
First available in Project Euclid: 3 March 2022

Digital Object Identifier: 10.3150/21-BEJ1366

Keywords: 35Q62 , 35R30 , 46L53 , 46L54 , 60B20 , 62G05 , 65M32

Rights: Copyright © 2022 ISI/BS


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Vol.28 • No. 2 • May 2022
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