Fokker–Planck equation

In statistical mechanics and information theory, the Fokker–Planck equation is a partial differential equation that describes the time evolution of the probability density function of the velocity of a particle under the influence of drag forces and random forces, as in Brownian motion. The equation can be generalized to other observables as well. The Fokker-Planck equation has multiple applications in information theory, graph theory, data science, finance, economics etc. It is named after Adriaan Fokker and Max Planck, who described it in 1914 and 1917. It is also known as the Kolmogorov forward equation, after Andrey Kolmogorov, who independently discovered it in 1931. When applied to particle position distributions, it is better known as the Smoluchowski equation (after Marian Smoluchowski), and in this context it is equivalent to the convection–diffusion equation. When applied to particle position and momentum distributions, it is known as the Klein–Kramers equation. The case with zero diffusion is the continuity equation. The Fokker–Planck equation is obtained from the master equation through Kramers–Moyal expansion. The first consistent microscopic derivation of the Fokker–Planck equation in the single scheme of classical and quantum mechanics was performed by Nikolay Bogoliubov and Nikolay Krylov. In one spatial dimension x, for an Itô process driven by the standard Wiener process and described by the stochastic differential equation (SDE) with drift and diffusion coefficient , the Fokker–Planck equation for the probability density of the random variable is In the following, use . Define the infinitesimal generator (the following can be found in Ref.): The transition probability , the probability of going from to , is introduced here; the expectation can be written as Now we replace in the definition of , multiply by and integrate over . The limit is taken on Note now that which is the Chapman–Kolmogorov theorem. Changing the dummy variable to , one gets which is a time derivative.