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Mat. Sb., 2021, Volume 212, Number 6, Pages 3–42 (Mi msb9427)  

On uniqueness of probability solutions of the Fokker-Planck-Kolmogorov equation

V. I. Bogachevab*, T. I. Krasovitskiiac, S. V. Shaposhnikovab

a Faculty of Mechanics and Mathematics, Lomonosov Moscow State University, Moscow, Russia
b National Research University Higher School of Economics, Moscow, Russia
c Moscow Center for Fundamental and Applied Mathematics, Moscow, Russia

Abstract: The paper gives a solution to the long-standing problem of uniqueness for probability solutions to the Cauchy problem for the Fokker-Planck-Kolmogorov equation with an unbounded drift coefficient and unit diffusion coefficient. It is proved that in the one-dimensional case uniqueness holds and in all other dimensions it fails. The case of nonconstant diffusion coefficients is also investigated.
Bibliography: 70 titles.

Keywords: Fokker-Planck-Kolmogorov equation, Cauchy problem, uniqueness problem.

Funding Agency Grant Number
Russian Foundation for Basic Research 20-01-00432-а
Moscow Center of Fundamental and Applied Mathematics
Foundation for the Advancement of Theoretical Physics and Mathematics BASIS 18-1-6-83-1
Simons Foundation
Contest «Young Russian Mathematics»
The research of V. I. Bogachev was carried out with the support of the Russian Foundation for Basic Research (grant no. 20-01-00432-a), the Moscow Center for Fundamental and Applied Mathematics, and the Foundation for the Advancement of Theoretical Physics and Mathematics “Basis” (grant no. 18-1-6-83-1). The research of T. I. Krasovitskii was carried out with the support of the Russian Foundation for Basic Research (grant no. 20-01-00432-a), the Moscow Center for Fundamental and Applied Mathematics, and a stipend from the Foundation for the Advancement of Theoretical Physics and Mathematics “Basis”. The research of S. V. Shaposhnikov was carried out with the support of the Russian Foundation for Basic Research (grant no. 20-01-00432-a), the Moscow Center for Fundamental and Applied Mathematics, the Simons-IUM fellowship, and a stipend from the contest “Young Russian Mathematics”.

* Author to whom correspondence should be addressed

DOI: https://doi.org/10.4213/sm9427

Full text: PDF file (735 kB)
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English version:
Sbornik: Mathematics, 2021, 212:6, 745–781

Bibliographic databases:

UDC: 517.955
MSC: 35Q84
Received: 21.04.2020 and 28.11.2020

Citation: V. I. Bogachev, T. I. Krasovitskii, S. V. Shaposhnikov, “On uniqueness of probability solutions of the Fokker-Planck-Kolmogorov equation”, Mat. Sb., 212:6 (2021), 3–42; Sb. Math., 212:6 (2021), 745–781

Citation in format AMSBIB
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\vol 212
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\pages 3--42
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