This thesis aims to prove a uniqueness theorem for the one dimensional driftless Fokker-Planck partial differential equation, i.e. the derivative in time direction of a probability density function p equals the second derivative in x (space) direction of the function a*p. Where the function 'a' is a strict positive Borel function with a boundary condition. We study two cases of the probability density function: In the first case its support is contained in the positive x- axis, in the second case the support is contained in the real numbers. The latter case was examined by M. Pierre. A schematic proof can be found in 'Peacocks and associated martingales with explicit constructions' by Hirsch, Profeta and Yor. We will replicate this proof but in more detail. However, the main result of this thesis is the first case. Unfortunately, the straightforward adaptation of the proof was not successful. An additional assumption for the function 'a' should fix this problem, a heuristical proof is shown.