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  4. On properties of solutions for Fokker-Planck-Kolmogorov equations

On properties of solutions for Fokker-Planck-Kolmogorov equations

MMC.
2020;
Volume 7, Number 1
: pp. 158–168
https://doi.org/10.23939/mmc2020.01.158
Received: March 17, 2020
Revised: April 20, 2020
Accepted: April 22, 2020

Mathematical Modeling and Computing, Vol. 7, No. 1, pp. 158–168 (2020)

Authors:
  1. I. P. Medynsky
1
Lviv Polytechnic National University

In the paper, we illuminate the connection between diffusion processes and partial differential equations of parabolic type.  The emphasis is on degenerate parabolic equations with real-valued coefficients.  These equations are the generalization of the classical Kolmogorov equation of diffusion with inertia, which may be treated as Fokker-Planck-Kolmogorov equations for the corresponding degenerate diffusion processes.  A fundamental solution of the Cauchy problem for Fokker-Planck-Kolmogorov equation determines the transition probabilities to the corresponding diffusion process. The conditions on the coefficients under which there exists the classical fundamental solution are formulated.  The basic properties of  fundamental solutions are proved.  The application of the fundamental solution to the investigation of correct solvability for the Cauchy problem is presented.

diffusion process
transition probability to a process
Fokker--Planck--Kolmogorov equation
degenerate parabolic equation
fundamental solution
Cauchy problem
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