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.
