Abstract
One proves the uniqueness of distributional solutions to nonlinear Fokker–Planck equations with monotone diffusion term and derive as a consequence (restricted) uniqueness in law for the corresponding McKean–Vlasov stochastic differential equation (SDE).
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17 November 2021
A Correction to this paper has been published: https://doi.org/10.1007/s40072-021-00223-9
References
Barbu, V.: Nonlinear Differential Equations of Monotone Type in Banach Spaces. Springer, New York (2010)
Barbu, V., Röckner, M.: From nonlinear Fokker-Planck equations to solutions of distribution dependent SDE. Ann. Probab. 48(4), 1902–1920 (2020)
Barbu, V., Röckner, M.: Probabilistic representation for solutions to nonlinear Fokker–Planck equation. SIAM J. Math. Anal. 50(4), 4246–4260 (2018)
Barbu, V., Röckner, M.: Solutions for nonlinear Fokker–Planck equations with measures as initial data and McKean–Vlasov equations. arXiv: 2005.02311 [math.PR]
Barbu, V., Röckner, M.: The evolution to equilibrium of solutions to nonlinear Fokker–Planck equations. arXiv:1904.082-91 [math.PR]
Barbu, V., Russo, F., Röckner, M.: Probabilistic representation for solutions of an irregular porous media type equation: the irregular degenerate case. Probab. Theory Rel. Fields 15, 1–43 (2011)
Belaribi, N., Russo, F.: Uniquness for Fokker–Planck equations with measurable coefficients and applications to the fast diffusion equations. Electron. J. Probab. 17, 1–28 (2012)
Brezis, H., Crandall, M.G.: Uniqueness of solutions of the initial-value problem for \(u_t-\Delta \beta (u)=0\). J. Math. Pures et Appl. 58, 153–163 (1979)
Pierre, M.: Uniqueness of the solutions of \(u_t-\Delta \varphi (u)=0\) with initial data measure. Nonlinear Anal. Theory Methods Appl. 6(2), 175–187 (1982)
Trevisan, D.: Well-posedness of multidimensional diffusion processes with weakly differentiable coefficients. Electron. J. Probab. 21, 22–41 (2016)
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This work was supported by the DFG through CRC 1283.
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Barbu, V., Röckner, M. Uniqueness for nonlinear Fokker–Planck equations and weak uniqueness for McKean–Vlasov SDEs. Stoch PDE: Anal Comp 9, 702–713 (2021). https://doi.org/10.1007/s40072-020-00181-8
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DOI: https://doi.org/10.1007/s40072-020-00181-8