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

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

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FokkerPlanck_HALarXiv.pdf (649Kb)
Type
Article accepté pour publication ou publié
Date
2022
Journal name
Bernoulli
Volume
28
Number
2
Publisher
International Statistical Institute
Pages
771-802
Publication identifier
10.3150/21-BEJ1366
Metadata
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Author(s)
Maïda, Mylène
Laboratoire Paul Painlevé [LPP]
Dat Nguyen, Tien
Laboratoire de Mathématiques d'Orsay [LMO]
Pham Ngoc, Thanh Mai
Laboratoire de Mathématiques d'Orsay [LMO]
Rivoirard, Vincent
CEntre de REcherches en MAthématiques de la DEcision [CEREMADE]
Tran, Viet-Chi
Laboratoire Analyse et Mathématiques Appliquées [LAMA]
Abstract (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 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.
Subjects / Keywords
PDE with random initial condition; free deconvolution; inverse problem; kernel estimation; Fourier transform; mean integrated square error; Dyson Brownian motion

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