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][3] In many textbooks, the expansion is used only to derive the Fokker–Planck equation, and never used again. In general, continuous stochastic processes are essentially all Markovian, and so Fokker–Planck equations are sufficient for studying them. The higher-order Kramers–Moyal expansion only come into play when the process is jumpy. This usually means it is a Poisson-like process.^[4][5]

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. This is implemented as a python package ^[6]

Statement

Start with the integro-differential master equation

${\displaystyle {\frac {\partial p(x,t)}{\partial t}}=\int p(x,t|x_{0},t_{0})p(x_{0},t_{0})dx_{0}}$

where ${\displaystyle p(x,t|x_{0},t_{0})}$ is the transition probability function, and ${\displaystyle p(x,t)}$ is the probability density at time ${\displaystyle t}$. The Kramers–Moyal expansion transforms the above to an infinite order partial differential equation^[7][8][9]

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

and also

$${\displaystyle \partial _{t}p(x,t|x_{0},t_{0})=\sum _{n=1}^{\infty }(-\partial _{x})^{n}}$$

where ${\displaystyle D_{n}(x,t)}$ are the Kramers–Moyal coefficients, defined by

$${\displaystyle D_{n}(x,t)={\frac {1}{n!}}\lim _{\tau \to 0}{\frac {1}{\tau }}\mu _{n}(t|x,t-\tau )}$$

${\displaystyle \mu _{n}}$

${\displaystyle \mu _{n}(t'|x,t)=\int _{-\infty }^{\infty }(x'-x)^{n}p(x',t'\mid x,t)\ dx'.}$

The Fokker–Planck equation is obtained by keeping only the first two terms of the series in which ${\displaystyle D_{1}}$ is the drift and ${\displaystyle D_{2}}$ is the diffusion coefficient.^[10]

Also, the moments, assuming they exist, evolves as^[11]

$${\displaystyle {\frac {\partial }{\partial t}}\left\langle x^{n}\right\rangle =\sum _{k=1}^{n}{\frac {n!}{(n-k)!}}\left\langle x^{n-k}D^{(k)}(x,t)\right\rangle }$$

${\displaystyle \left\langle f\right\rangle =\int f(x)p(x,t)dx}$

n-dimensional version

The above version is the one-dimensional version. It generalizes to n-dimensions. (Section 4.7 ^[9])

Proof

In usual probability, where the probability density does not change, the moments of a probability density function determines the probability density itself by a Fourier transform (details may be found at the characteristic function page):

$${\displaystyle p(x)={\frac {1}{2\pi }}\int e^{-ikx}{\tilde {p}}(k)dk=\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{n!}}\delta ^{(n)}(x)\mu _{n}}$$