The Ornstein-Uhlenbeck equation as a limiting case of a successive interactions model

A model for the motion of a test particle through a medium composed of many other particles is presented. It supposes that the test particle interacts with another particle up to a random time U₁, and then another particle up to a random time U₂, and so on. The test particle-particle interactions end when either another particle interrupts the interaction or the impulse of the force due to the current interaction exceeds a given bound. The 'random walk on polynomials' process is introduced in this paper and is used to model the total impulse of the force arising from these interactions. It is shown that as the duration of these interactions approaches zero, the velocity of the test particle becomes a solution of an Ornstein-Uhlenbeck equation. The renormalization required for this convergence shows that the limiting dynamics involve interactions of short duration and relatively large forces, as well as a relatively massive test particle.