Kramers–Moyal expansion

In stochastic processes, Kramers–Moyal expansion refers to a Taylor series expansion of the master equation, named after Hans Kramers and José Enrique Moyal.^[1][2] This expansion transforms the integro-differential master equation

${\displaystyle {\frac {\partial p(x,t)}{\partial t}}=\int dx'[W(x|x')p(x',t)-W(x'|x)p(x,t)]}$

where ${\displaystyle p(x,t|x_{0},t_{0})}$ (for brevity, this probability is denoted by ${\displaystyle p(x,t)}$) is the transition probability density, to an infinite order partial differential equation^[3][4][5]

${\displaystyle {\frac {\partial p(x,t)}{\partial t}}=\sum _{n=1}^{\infty }{\frac {(-1)^{n}}{n!}}{\frac {\partial ^{n}}{\partial x^{n}}}[\alpha _{n}(x)p(x,t)]}$

where

${\displaystyle \alpha _{n}(x)=\int _{-\infty }^{\infty }(x'-x)^{n}W(x'\mid x)\ dx'.}$

Here ${\displaystyle W(x'\mid x)}$ is the transition probability rate. The Fokker–Planck equation is obtained by keeping only the first two terms of the series in which ${\displaystyle \alpha _{1}}$ is the drift and ${\displaystyle \alpha _{2}}$ is the diffusion coefficient.

Pawula theorem[]

The Pawula theorem states that the expansion either stops after the first term or the second term.^[6][7] If the expansion continues past the second term it must contain an infinite number of terms, in order that the solution to the equation be interpretable as a probability density function.^[8]

Implementations[]

• Implementation as a python package ^[9]

References[]