On properties of solutions for Fokker-Planck-Kolmogorov equations

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.

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