Derivation of a Fokker-Planck Equation Drift and Diffusion of a Probability Density in State Space

To derive a Fokker-Planck equation, which deals with probabilities, we should start by stating the probabilistic notation and properties we will use. The probability that event A occurs is P [A]. Summing over all possible mutually exclusive events must yield a total of one: ∑ A P [A] = 1. If you include non-exlusive events, the probability sum can exceed one, so any sums will be performed over mutually exclusive events. When summations are not involved, the events named can be partially or fully simultaneous, allowing illustration using Venn diagrams with overlapping regions. The probability that both A and B occur is a joint probability P [A,B]. The probability that A occurs given that B occurs is a conditional probability P [A|B]. A joint-conditional probability property follows: