Kramers–Moyal expansion
In stochastic processes, the Kramers–Moyal expansion refers to a Taylor series expansion of the master equation, and is named after Hans Kramers and José Enrique Moyal. In many textbooks, the expansion is only used to derive the Fokker–Planck equation, and never used again. In general, continuous stochastic processes are essentially Markovian, and so Fokker–Planck equations are sufficient for studying them. The higher-order Kramers–Moyal expansion only comes 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.
Statement
Start with the integro-differential master equationwhere is the transition probability function, and is the probability density at time. The Kramers–Moyal expansion transforms the above to an infinite order partial differential equation
and also
where are the Kramers–Moyal coefficients, defined byand are the central moment functions, defined by
The Fokker–Planck equation is obtained by keeping only the first two terms of the series in which is the drift and is the diffusion coefficient.
The moments, assuming they exist, evolve as
where angled brackets mean taking the expectation:.
n-dimensional version
The above is the one-dimensional version. It generalizes to n-dimensions.Proof
In usual probability, where the probability density does not change, the moments of a probability density function determine the probability density itself by a Fourier transform :Similarly,Now we need to integrate away the Dirac delta function. Fixing a small, we have by the Chapman-Kolmogorov equation,The term is just, so taking derivative with respect to time,
The same computation with gives the other equation.
Forward and backward equations
The equation can be recast into a linear operator form, using the idea of an infinitesimal generator. Define the linear operator then the equation above states that In this form, the equations are precisely in the form of a general Kolmogorov forward equation. The backward equation then states thatwhereis the Hermitian adjoint of.
Computing the Kramers–Moyal coefficients
By definition,This definition works because, as those are the central moments of the Dirac delta function. Since the even central moments are nonnegative, we have for all. When the stochastic process is the Markov process, we can directly solve for as approximated by a normal distribution with mean and variance. This allows us to compute the central moments, and soThis then gives us the 1-dimensional Fokker–Planck equation:Pawula theorem
The Pawula theorem states that either the sequence becomes zero at the third term, or all its even terms are positive.Proof
By the Cauchy–Schwarz inequality, the central moment functions satisfy. So, taking the limit, we have. If some for some, then. In particular,. So the existence of any nonzero coefficient of order implies the existence of nonzero coefficients of arbitrarily large order. Also, if, then. So the existence of any nonzero coefficient of order implies all coefficients of order are positive.Interpretation
Let the operator be defined such that. The probability density evolves by. A different order of gives a different level of approximation.- : the probability density does not evolve
- : it evolves by deterministic drift only.
- : it evolves by drift and Brownian motion
- : the fully exact equation.