On a randomly accelerated particle

Abstract

The focus of this note is to learn more about the Kolmogorov equation describing the dynamics of a randomly accelerated particle. We first explore some existing results of the Kolmogorov equation from the stochastic and differential equation points of view and discuss its solvability with and without boundary conditions. More specifically, we introduce stochastic processes and Brownian motion and we present a connection between a stochastic process and a differential equation. After looking at stochastic processes, we introduce generalized functions and derive the fundamental solution to the heat equation and to the Fokker–Planck equation. The problem with a reflecting boundary condition is also studied by using various methods such as separation of variables, self-similarity, and the reflection method.