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July 2020 From nonlinear Fokker–Planck equations to solutions of distribution dependent SDE
Viorel Barbu, Michael Röckner
Ann. Probab. 48(4): 1902-1920 (July 2020). DOI: 10.1214/19-AOP1410

Abstract

We construct weak solutions to the McKean–Vlasov SDE \begin{equation*}dX(t)=b\biggl(X(t),\frac{d{\mathcal{L}}_{X(t)}}{dx}\bigl(X(t)\bigr)\biggr)\,dt+\sigma\biggl(X(t),\frac{d{\mathcal{L}}_{X(t)}}{dt}\bigl(X(t)\bigr)\biggr)\,dW(t)\end{equation*} on ${\mathbb{R}}^{d}$ for possibly degenerate diffusion matrices $\sigma$ with $X(0)$ having a given law, which has a density with respect to Lebesgue measure, $dx$. Here, ${\mathcal{L}}_{X(t)}$ denotes the law of $X(t)$. Our approach is to first solve the corresponding nonlinear Fokker–Planck equations and then use the well-known superposition principle to obtain weak solutions of the above SDE.

Citation

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Viorel Barbu. Michael Röckner. "From nonlinear Fokker–Planck equations to solutions of distribution dependent SDE." Ann. Probab. 48 (4) 1902 - 1920, July 2020. https://doi.org/10.1214/19-AOP1410

Information

Received: 1 September 2018; Revised: 1 August 2019; Published: July 2020
First available in Project Euclid: 20 July 2020

zbMATH: 07224963
MathSciNet: MR4124528
Digital Object Identifier: 10.1214/19-AOP1410

Subjects:
Primary: 60G46, 60H10, 60H30
Secondary: 35C99, 58J165

Rights: Copyright © 2020 Institute of Mathematical Statistics

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Vol.48 • No. 4 • July 2020
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