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]
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):
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.
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 then allows us to compute the central moments, and so
This then gives us the 1-dimensional Fokker–Planck equation:
By 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.
Let the operator be defined such . The probability density evolves by . Different order of gives different level of approximation.
: the probability density does not evolve
: it evolves by deterministic drift only.
: it evolves by drift and Brownian motion (Fokker-Planck equation)
: the fully exact equation.
Pawula theorem means that if truncating to the second term is not exact, that is, , then truncating to any term is still not exact. Usually, this means that for any truncation , there exists a probability density function that can become negative during its evolution (and thus fail to be a probability density function). However, this doesn't mean that Kramers-Moyal expansions truncated at other choices of 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 approximation.
^Spinney, Richard; Ford, Ian (2013). "Fluctuation relations: a pedagogical overview". In Klages, Rainer; Just, Wolfram; Jarzynski, Christopher (eds.). Nonequilibrium Statistical Physics of Small Systems: Fluctuation relations and beyond. Reviews of Nonlinear Dynamics and Complexity. Weinheim: Wiley-VCH. pp. 3–56. arXiv:1201.6381. doi:10.1002/9783527658701.ch1. ISBN978-3-527-41094-1. MR3308060.
^Gardiner, C. (2009). Stochastic Methods (4th ed.). Berlin: Springer. ISBN978-3-642-08962-6.
^Van Kampen, N. G. (1992). Stochastic Processes in Physics and Chemistry. Elsevier. ISBN0-444-89349-0.
^ abRisken, H. (1996). The Fokker–Planck Equation. Berlin, Heidelberg: Springer. pp. 63–95. ISBN3-540-61530-X.
^Paul, Wolfgang; Baschnagel, Jörg (2013). "A Brief Survey of the Mathematics of Probability Theory". Stochastic Processes. Springer. pp. 17–61 [esp. 33–35]. doi:10.1007/978-3-319-00327-6_2.