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]

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]

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

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

where ${\displaystyle D_{n}(x,t)}$ are the Kramers–Moyal coefficients, defined by$${\displaystyle D_{n}(x,t)={\frac {1}{n!}}\lim _{\tau \to 0}{\frac {1}{\tau }}\mu _{n}(t|x,t-\tau )}$$and ${\displaystyle \mu _{n}}$ are the central moment functions, defined by

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

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]

$${\displaystyle {\frac {\partial }{\partial t}}\left\langle x^{n}\right\rangle =\sum _{k=1}^{n}{\frac {n!}{(n-k)!}}\left\langle x^{n-k}D^{(k)}(x,t)\right\rangle }$$where angled brackets mean taking the expectation: ${\displaystyle \left\langle f\right\rangle =\int f(x)p(x,t)dx}$.

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

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):$${\displaystyle p(x)={\frac {1}{2\pi }}\int e^{-ikx}{\tilde {p}}(k)dk=\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{n!}}\delta ^{(n)}(x)\mu _{n}}$$$${\displaystyle {\tilde {p}}(k)=\int e^{ikx}p(x)dx=\sum _{n=0}^{\infty }{\frac {(ik)^{n}}{n!}}\mu _{n}}$$Similarly, $${\displaystyle p(x,t|x_{0},t_{0})=\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{n!}}\delta ^{(n)}(x-x_{0})\mu _{n}(t|x_{0},t_{0})}$$ Now we need to integrate away the Dirac delta function. Fixing a small ${\displaystyle \tau >0}$, we have by the Chapman-Kolmogorov equation,$${\displaystyle {\begin{aligned}p(x,t)&=\int p(x,t|x',t-\tau )p(x',t-\tau )dx'\\&=\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{n!}}\int p(x',t-\tau )\delta ^{(n)}(x-x')\mu _{n}(t|x',t-\tau )dx'\\&=\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{n!}}\partial _{x}^{n}(p(x,t-\tau )\mu _{n}(t|x,t-\tau ))\end{aligned}}}$$The ${\displaystyle n=0}$ term is just ${\displaystyle p(x,t-\tau )}$, so taking derivative with respect to time,$${\displaystyle \partial _{t}p(x,t)=\lim _{\tau \to 0^{+}}{\frac {1}{\tau }}\sum _{n=1}^{\infty }{\frac {(-1)^{n}}{n!}}\partial _{x}^{n}(p(x,t-\tau )\mu _{n}(t|x,t-\tau ))=\sum _{n=1}^{\infty }(-\partial _{x})^{n}(p(x,t)D_{n}(x,t))}$$

The same computation with ${\displaystyle p(x,t|x_{0},t_{0})}$ gives the other equation.

The equation can be recast into a linear operator form, using the idea of infinitesimal generator. Define the linear operator $${\displaystyle {\mathcal {A}}f:=\sum _{n=1}^{\infty }(-\partial _{x})^{n}[D_{n}(x,t)f(x,t)]}$$then the equation above states $${\displaystyle {\begin{aligned}\partial _{t}p(x,t)&={\mathcal {A}}p(x,t)\\\partial _{t}p(x,t|x_{0},t_{0})&={\mathcal {A}}p(x,t|x_{0},t_{0})\end{aligned}}}$$In this form, the equations are precisely in the form of a general Kolmogorov forward equation. The backward equation then states that$${\displaystyle \partial _{t}p(x_{1},t_{1}|x,t)=-{\mathcal {A}}^{\dagger }p(x_{1},t_{1}|x,t)}$$where$${\displaystyle {\mathcal {A}}^{\dagger }f:=\sum _{n=1}^{\infty }D_{n}(x,t)\partial _{x}^{n}[f(x,t)]}$$ is the Hermitian adjoint of ${\displaystyle {\mathcal {A}}}$.

By definition,$${\displaystyle D_{n}(x,t)={\frac {1}{n!}}\lim _{\tau \to 0}{\frac {1}{\tau }}\mu _{n}(t|x,t-\tau )}$$This definition works because ${\displaystyle \mu _{n}(t|x,t)=0}$, as those are the central moments of the Dirac delta function. Since the even central moments are nonnegative, we have ${\displaystyle D_{2n}\geq 0}$ for all ${\displaystyle n\geq 1}$. When the stochastic process is the Markov process ${\displaystyle dX=bdt+\sigma dW_{t}}$, we can directly solve for ${\displaystyle p(x,t|x,t-\tau )}$ as approximated by a normal distribution with mean ${\displaystyle x+b(x)\tau }$ and variance ${\displaystyle \sigma ^{2}\tau }$. This then allows us to compute the central moments, and so$${\displaystyle D_{1}=b,\quad D_{2}={\frac {1}{2}}\sigma ^{2},\quad D_{3}=D_{4}=\cdots =0}$$This then gives us the 1-dimensional Fokker–Planck equation:$${\displaystyle \partial _{t}p=-\partial _{x}(bp)+{\frac {1}{2}}\partial _{x}^{2}(\sigma ^{2}p)}$$

Pawula theorem states that either the sequence ${\displaystyle D_{1},D_{2},D_{3},...}$ becomes zero at the third term, or all its even terms are positive.^[12][13]

By Cauchy–Schwarz inequality, the central moment functions satisfy ${\displaystyle \mu _{n+m}^{2}\leq \mu _{2n}\mu _{2m}}$. So, taking the limit, we have ${\displaystyle D_{n+m}^{2}\leq {\frac {(2n)!(2m)!}{(n+m)!^{2}}}D_{2n}D_{2m}}$. If some ${\displaystyle D_{2+n}\neq 0}$ for some ${\displaystyle n\geq 1}$, then ${\displaystyle D_{2}D_{2+2n}>0}$. In particular, ${\displaystyle D_{2+n},D_{2+2n},D_{2+4n},...>0}$. So the existence of any nonzero coefficient of order ${\displaystyle \geq 3}$ implies the existence of nonzero coefficients of arbitrarily large order. Also, if ${\displaystyle D_{n}\neq 0}$, then ${\displaystyle D_{2}D_{2n-2}>0,D_{4}D_{2n-4}>0,...}$. So the existence of any nonzero coefficient of order ${\displaystyle n}$ implies all coefficients of order ${\displaystyle 2,4,...,2n-2}$ are positive.

Let the operator ${\displaystyle {\mathcal {A}}_{m}}$ be defined such ${\displaystyle {\mathcal {A}}_{m}f:=\sum _{n=1}^{m}(-\partial _{x})^{n}[D_{n}(x,t)f(x,t)]}$. The probability density evolves by ${\displaystyle \partial _{t}\rho \approx {\mathcal {A}}_{m}\rho }$. Different order of ${\displaystyle m}$ gives different level of approximation.

• ${\displaystyle m=0}$: the probability density does not evolve
• ${\displaystyle m=1}$: it evolves by deterministic drift only.
• ${\displaystyle m=2}$: it evolves by drift and Brownian motion (Fokker-Planck equation)
• ${\displaystyle m=\infty }$: the fully exact equation.

Pawula theorem means that if truncating to the second term is not exact, that is, ${\displaystyle {\mathcal {A}}_{2}\neq {\mathcal {A}}}$, then truncating to any term is still not exact. Usually, this means that for any truncation ${\displaystyle {\mathcal {A}}_{m}}$, there exists a probability density function ${\displaystyle \rho }$ that can become negative during its evolution ${\displaystyle \partial _{t}\rho \approx {\mathcal {A}}_{m}\rho }$ (and thus fail to be a probability density function). However, this doesn't mean that Kramers-Moyal expansions truncated at other choices of ${\displaystyle m}$ is useless. Though the solution must have negative values at least for sufficiently small times, the resulting approximation probability density may still be better than the ${\displaystyle m=2}$ approximation.