diffusion process

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.