Geodesic
On nonuniqueness of probability solutions to the Cauchy problem for the Fokker--Planck--Kolmogorov equation
Doklady Rossijskoj akademii nauk. Matematika, informatika, processy upravleniâ, Tome 498 (2021), pp. 16-20

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In this paper we give a positive answer to the question about the possibility of existence of several probability solutions to the Fokker–Planck–Kolmogorov equation for all initial conditions: we construct the first example of an equation with a unit diffusion matrix and a smooth drift coefficient for which the Cauchy problem with every probability initial condition has an infinite-dimensiona1 simplex of probability solutions.
Keywords: Fokker–Planck–Kolmogorov equation, Cauchy problem, uniqueness of a probability solution. 
@article{DANMA_2021_498_a2,
     author = {V. I. Bogachev and T. I. Krasovitskii and S. V. Shaposhnikov},
     title = {On nonuniqueness of probability solutions to the {Cauchy} problem for the {Fokker--Planck--Kolmogorov} equation},
     journal = {Doklady Rossijskoj akademii nauk. Matematika, informatika, processy upravleni\^a},
     pages = {16--20},
     publisher = {mathdoc},
     volume = {498},
     year = {2021},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/DANMA_2021_498_a2/}
}
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V. I. Bogachev; T. I. Krasovitskii; S. V. Shaposhnikov. On nonuniqueness of probability solutions to the Cauchy problem for the Fokker--Planck--Kolmogorov equation. Doklady Rossijskoj akademii nauk. Matematika, informatika, processy upravleniâ, Tome 498 (2021), pp. 16-20. http://geodesic.mathdoc.fr/item/DANMA_2021_498_a2/