Abstract:
We consider the Cauchy problem for the Fokker–Planck–Kolmogorov equation where the coefficients and the initial data depend measurably on a parameter and show that there is a selection of a solution measurable with respect to the parameter. In particular, the coefficients and the initial data themselves can be regarded as parameters. Several settings of the problem are discussed in which the measurability of solutions with respect to parameters is established.
Keywords:Fokker–Planck–Kolmogorov equation, Cauchy problem, measurability with respect to a parameter.
The work of the first and third authors (presented in Secs. 2 and 3 was accomplished with the support of the Ministry of Education and Science of the Russian Federation as part of the program of the Moscow
Center for Fundamental and Applied Mathematics under the agreement no. 075-15-2022-284,
The work of the second author (presented in Sec. 4) was conducted within
the framework of the HSE University Basic Research Program.
Citation:
V. I. Bogachev, I. I. Malofeev, S. V. Shaposhnikov, “On dependence of solutions to Fokker–Planck–Kolmogorov equations on their coefficients and initial data”, Math. Notes, 116:3 (2024), 421–431
\Bibitem{BogMalSha24}
\by V.~I.~Bogachev, I.~I.~Malofeev, S.~V.~Shaposhnikov
\paper On dependence of solutions to Fokker--Planck--Kolmogorov equations on their coefficients and initial data
\jour Math. Notes
\yr 2024
\vol 116
\issue 3
\pages 421--431
\mathnet{http://mi.mathnet.ru/mzm14489}
\crossref{https://doi.org/10.1134/S0001434624090037}