Kramers–Moyal expansion

In stochastic processes, the Kramers–Moyal expansion refers to a Taylor series expansion of the master equation, and is named after Hans Kramers and José Enrique Moyal. In many textbooks, the expansion is only used to derive the Fokker–Planck equation, and never used again. In general, continuous stochastic processes are essentially Markovian, and so Fokker–Planck equations are sufficient for studying them. The higher-order Kramers–Moyal expansion only comes into play when the process is jumpy. This usually means it is a Poisson-like process.

For a real stochastic process, one can compute its central-moment functions from experimental data on the process, from which one can then compute its Kramers–Moyal coefficients, and thus empirically measure its Kolmogorov forward and backward equations.