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Doklady Rossijskoj Akademii Nauk. Mathematika, Informatika, Processy Upravlenia, 2021, Volume 498, Pages 16–20
DOI: https://doi.org/10.31857/S2686954321030048 (Mi danma170) 		 
This article is cited in 1 scientific paper (total in 1 paper)

MATHEMATICS

On nonuniqueness of probability solutions to the Cauchy problem for the Fokker–Planck–Kolmogorov equation

V. I. Bogachevabcd, T. I. Krasovitskiiac, S. V. Shaposhnikovabc

a 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
d St. Tikhon's Orthodox University, Moscow, Russia
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DOI: https://doi.org/10.31857/S2686954321030048
Abstract: 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.
Funding agency Grant number
Russian Foundation for Basic Research 20-01-00432
Simons Foundation
Contest «Young Russian Mathematics»
This research was supported by the Russian Foundation for Basic Research (grant no. 20-01-00432) and the Moscow Center for Fundamental and Applied Mathematics. The third author acknowledges the support of the Simons Foundation; he is a winner of the “Young Russian Mathematics” contest and thanks its sponsors and jury.
Presented: A. N. Shiryaev
Received: 26.03.2021
Revised: 26.03.2021
Accepted: 04.04.2021
English version:
Doklady Mathematics, 2021, Volume 103, Issue 3, Pages 108–112
DOI: https://doi.org/10.1134/S1064562421030042
Bibliographic databases:
Document Type: Article
UDC: 517.955
Language: Russian
Citation: V. I. Bogachev, T. I. Krasovitskii, S. V. Shaposhnikov, “On nonuniqueness of probability solutions to the Cauchy problem for the Fokker–Planck–Kolmogorov equation”, Dokl. RAN. Math. Inf. Proc. Upr., 498 (2021), 16–20; Dokl. Math., 103:3 (2021), 108–112
Citation in format AMSBIB
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\by V.~I.~Bogachev, T.~I.~Krasovitskii, S.~V.~Shaposhnikov
\paper On nonuniqueness of probability solutions to the Cauchy problem for the Fokker--Planck--Kolmogorov equation
\jour Dokl. RAN. Math. Inf. Proc. Upr.
\yr 2021
\vol 498
\pages 16--20
\mathnet{http://mi.mathnet.ru/danma170}
\crossref{https://doi.org/10.31857/S2686954321030048}
\zmath{https://zbmath.org/?q=an:7424711}
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\transl
\jour Dokl. Math.
\yr 2021
\vol 103
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\pages 108--112
\crossref{https://doi.org/10.1134/S1064562421030042}
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