In stochastic processes, Kramers–Moyal expansion refers to a Taylor series expansion of the master equation, named after Hans Kramers and José Enrique Moyal. 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.
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 
Statement.
Start with the integro-differential master equation 
formula_1
where formula_2 is the transition probability function, and formula_3 is the probability density at time formula_4. The Kramers–Moyal expansion transforms the above to an infinite order partial differential equation
formula_5
and alsoformula_6
where formula_7 are the Kramers–Moyal coefficients, defined byformula_8and formula_9 are the central moment functions, defined by
formula_10
The Fokker–Planck equation is obtained by keeping only the first two terms of the series in which formula_11 is the drift and formula_12 is the diffusion coefficient.
Also, the moments, assuming they exist, evolves as
formula_13where angled brackets mean taking the expectation: formula_14.
n-dimensional version.
The above version is the one-dimensional version. It generalizes to n-dimensions. (Section 4.7 )
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):formula_15formula_16Similarly, 
formula_17
Now we need to integrate away the Dirac delta function. Fixing a small formula_18, we have by the Chapman-Kolmogorov equation,formula_19The formula_20 term is just formula_21, so taking derivative with respect to time,formula_22
The same computation with formula_2 gives the other equation.
Forward and backward equations.
The equation can be recast into a linear operator form, using the idea of infinitesimal generator. Define the linear operator formula_24then the equation above states formula_25In this form, the equations are precisely in the form of a general Kolmogorov forward equation. The backward equation then states thatformula_26whereformula_27
is the Hermitian adjoint of formula_28.
Computing the Kramers–Moyal coefficients.
By definition,formula_8This definition works because formula_30, as those are the central moments of the Dirac delta function. Since the even central moments are nonnegative, we have formula_31 for all formula_32. When the stochastic process is the Markov process formula_33, we can directly solve for formula_34 as approximated by a normal distribution with mean formula_35 and variance formula_36. This then allows us to compute the central moments, and soformula_37This then gives us the 1-dimensional Fokker–Planck equation:formula_38
Pawula theorem.
Pawula theorem states that either the sequence formula_39 becomes zero at the third term, or all its even terms are positive.
Proof.
By Cauchy–Schwarz inequality, the central moment functions satisfy formula_40. So, taking the limit, we have formula_41. If some formula_42 for some formula_43, then formula_44. In particular, formula_45. So the existence of any nonzero coefficient of order formula_46 implies the existence of nonzero coefficients of arbitrarily large order. Also, if formula_47, then formula_48. So the existence of any nonzero coefficient of order formula_49 implies all coefficients of order formula_50 are positive. 
Interpretation.
Let the operator formula_51 be defined such formula_52. The probability density evolves by formula_53. Different order of formula_54 gives different level of approximation.
Pawula theorem means that if truncating to the second term is not exact, that is, formula_59, then truncating to any term is still not exact. Usually, this means that for any truncation formula_60, there exists a probability density function formula_61 that can become negative during its evolution formula_62 (and thus fail to be a probability density function). However, this doesn't mean that Kramers-Moyal expansions truncated at other choices of formula_54 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 formula_57 approximation.
References.
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