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We prove a generalization of the known result of Trevisan on the Ambrosio-Figalli-Trevisan superposition principle for probability solutions to the Cauchy problem for the Fokker-Planck-Kolmogorov equation, according to which such a solution is generated by a solution to the corresponding martingale problem.
Ambrosio L
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Bogachev V.I., Krylov N.V., Röckner M. Elliptic and parabolic equations for measures. Uspehi Matem. Nauk. 2009. V. 64, N 6. P. 5–116 (in Russian); English transl.: Russian Math. Surveys. 2009. V. 64, N 6. P. 973–1078
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Bogachev V.I., Da Prato G., Röckner M. Existence and uniqueness of solutions for Fokker–Planck equations on Hilbert spaces. J. Evol. Equ. 10 (2010), no. 3, 487–509
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Bogachev V.I., Da Prato G., Röckner M. Uniqueness for solutions of Fokker–Planck equations on infinite dimensional spaces. Comm. Partial Differential Equations 36 (2011), no. 6, 925–939
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Bogachev V.I., Röckner M., Shaposhnikov S.V. On uniqueness problems related to elliptic equations for measures. J. Math. Sci. (New York). 2011. V. 176, N 6. P. 759–773
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Bogachev V.I., Röckner M., Shaposhnikov S.V. On uniqueness of solutions to the Cauchy problem for degenerate Fokker–Planck–Kolmogorov equations. J. Evol. Equat., V. 13, N 3, P. 577–593 (2013)
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Manita O.A., Shaposhnikov S.V. On the Cauchy problem for Fokker–Planck–Kolmogorov equations with potential terms on arbitrary domains. J. Dynamics Differ. Equ. 28 (2016), 493–518
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Trevisan, D. Well-posedness of multidimensional diffusion processes with weakly differentiable coefficients. Electron. J. Probab. 21 (2016), Paper No. 22, 41 pp
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Ambrosio, L., Trevisan, D. Well-posedness of Lagrangian flows and continuity equations in metric measure spaces, Anal. PDE 7 (2014), no. 5, 1179–1234
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Manita O.A., Shaposhnikov S.V. Nonlinear parabolic equations for measures. Algebra i Analiz 25 (2013), no. 1, 64–93 (in Russian); English transl.: St. Petersburg Math. J. 25 (2014), no. 1, 43–62
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Bogachev V.I., Da Prato G., Röckner M., Shaposhnikov S.V. An analytic approach to infinite-dimensional continuity and Fokker–Planck–Kolmogorov equations. Annali Scuola Norm. Pisa. V. 14, 983–1023 (2015)
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Bogachev V.I., Krylov N.V., Röckner M., Shaposhnikov S.V. Fokker–Planck–Kolmogorov equations. Amer. Math. Soc., Rhode Island, Providence, 2015
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Bogachev V.I. , Roeckner M., Shaposhnikov S.V. Uniqueness problems for degenerate Fokker–Planck–Kolmogorov equations. J. Math. Sci. (New York), V. 207, N 2, 147–165 (2015)
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Manita O.A., Romanov, M.S., Shaposhnikov S.V. On uniqueness of solutions to nonlinear Fokker–Planck–Kolmogorov equations. Nonlinear Anal. 128 (2015), 199–226
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Barbu V., Röckner M. Probabilistic representation for solutions to nonlinear Fokker–Planck equations. SIAM J. Math. Anal. 50 (2018), no. 4, 4246–4260
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Bogachev V.I. Weak convergence of measures. Amer. Math. Soc., Rhode Island, Providence, 2018
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Luo, D. The Itô SDEs and Fokker–Planck equations with Osgood and Sobolev coefficients. Stochastics 90 (2018), no. 3, 379–410
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