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JOURNALS // Doklady Rossijskoj Akademii Nauk. Mathematika, Informatika, Processy Upravlenia // Archive

Dokl. RAN. Math. Inf. Proc. Upr., 2021 Volume 498, Pages 16–20 (Mi danma170)

This article is cited 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

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.

UDC: 517.955

Presented: A. N. Shiryaev
Received: 26.03.2021
Revised: 26.03.2021
Accepted: 04.04.2021

DOI: 10.31857/S2686954321030048


 English version:
Doklady Mathematics, 2021, 103:3, 108–112

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