Deviation inequallities for exponential jump-diffusion processes

We clarify the connection between diffusion processes and partial differential equations of the parabolic type. The emphasis is on degenerate parabolic equations. These equations are a generalization of the classical Kolmogorov equation of diffusion with inertia which may be treated as the Fokker-Planck-Kolmogorov equations for the respectively degenerate diffusion processes. The basic results relating to the fundamental solution and the correct solvability of the Cauchy problem are presented.