On the superposition principle for probability solutions to Fokker-Planck-Kolmogorov equations

The superposition principle expresses a deep connection between the solutions of the martingale problem and the probability solutions of the Fokker-Planck-Kolmogorov equation. It has been extensively studied in recent years and the best-known results were obtained by L. Ambrosio, A. Figalli, and D. Trevisan. We will present a generalization of the superposition principle in the case of unbounded coefficients and arbitrary domain, demonstrate several counterexamples and formulate open problems.