On the Equation of Diffusion

Under the title "The Analytical Theory of Probability," A. Kolmogoroff [1] published a few years ago a mathematically elegant account of what might be called the general theory of diffusion. As the main result of his analysis, he sets up two differential equations for transition probabilities (see below). The first of them contains as its independent variables the initial parameters of the transition probability and has an analytical form resembling the ordinary diffusion equation. Unlike it, the second differential equation (depending on the final parameters) has a different form unfamiliar to the physicist. The puzzling asymmetry of the two equations, as to whose physical reasons the author gives no hint, was, in our opinion, the main cause why the work of Kolmogoroff did not receive more attention. Unless these reasons are fully understood, an intelligent application of the theory to practical problems is very difficult. We think it, therefore, worth while to present in the following lines a discussion of the mutual connection between the two equations of Kolmogoroff and of their relation to other forms of the diffusion equation used in physics.