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. This expansion transforms the integro-differential master equation

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

where $p(x,t|x_{0},t_{0})$ (for brevity, this probability is denoted by $p(x,t)$) is the transition probability density, to an infinite order partial differential equation

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

where

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

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

Pawula theorem

The Pawula theorem states that the expansion either stops after the first term or the second term. 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.

Implementations

• Implementation as a python package