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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] This expansion transforms the integro-differential master equation

{\frac {\partial p(x,t)}{\partial t}}=\int dx'[W(x|x')p(x',t)-W(x'|x)p(x,t)]

where $p(x,t|x_{0},t_{0})$ (for brevity, this probability is denoted by $p(x,t)$ ) is the transition probability density, to an infinite order partial differential equation^[3]^[4]^[5]

{\frac {\partial p(x,t)}{\partial t}}=\sum _{n=1}^{\infty }{\frac {(-1)^{n}}{n!}}{\frac {\partial ^{n}}{\partial x^{n}}}[\alpha _{n}(x)p(x,t)]

where

\alpha _{n}(x)=\int _{-\infty }^{\infty }(x'-x)^{n}W(x'\mid x)\ dx'.

Here $W(x'\mid x)$ is the transition probability rate. The Fokker–Planck equation is obtained by keeping only the first two terms of the series in which $\alpha _{1}$ is the drift and $\alpha _{2}$ is the diffusion coefficient.^[6]

Pawula theorem

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

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

Pawula theorem states that for any other choice of $m$ , there exists a probability density function $\rho$ that can become negative during its evolution $\partial _{t}\rho \approx L_{m}\rho$ (and thus fail to be a probability density function).^[7]^[8]^[9]

However, this doesn't mean that Kramers-Moyal expansions truncated at other choices of $m$ 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 $m=2$ approximation.

Implementations

Implementation as a python package ^[10]

References

^ Kramers, H. A. (1940). "Brownian motion in a field of force and the diffusion model of chemical reactions". Physica. 7 (4): 284–304. Bibcode:1940Phy.....7..284K. doi:10.1016/S0031-8914(40)90098-2. S2CID 33337019.
^ Moyal, J. E. (1949). "Stochastic processes and statistical physics". Journal of the Royal Statistical Society. Series B (Methodological). 11 (2): 150–210. JSTOR 2984076.
^ Gardiner, C. (2009). Stochastic Methods (4th ed.). Berlin: Springer. ISBN 978-3-642-08962-6.
^ Van Kampen, N. G. (1992). Stochastic Processes in Physics and Chemistry. Elsevier. ISBN 0-444-89349-0.
^ Risken, H. (1996). The Fokker–Planck Equation. Berlin, Heidelberg: Springer. pp. 63–95. ISBN 3-540-61530-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.
^ Pawula, R. F. (1967). "Generalizations and extensions of the Fokker–Planck–Kolmogorov equations" (PDF). IEEE Transactions on Information Theory. 13 (1): 33–41. doi:10.1109/TIT.1967.1053955.
^ Pawula, R. F. (1967). "Approximation of the linear Boltzmann equation by the Fokker–Planck equation". Physical Review. 162 (1): 186–188. Bibcode:1967PhRv..162..186P. doi:10.1103/PhysRev.162.186.
^ Risken, Hannes (6 December 2012). The Fokker-Planck Equation: Methods of Solution and Applications. ISBN 9783642968075.
^ Rydin Gorjão, L.; Meirinhos, F. (2019). "kramersmoyal: Kramers--Moyal coefficients for stochastic processes". Journal of Open Source Software. 4 (44): 1693. arXiv:1912.09737. Bibcode:2019JOSS....4.1693G. doi:10.21105/joss.01693.

[1] Kramers, H. A. (1940). "Brownian motion in a field of force and the diffusion model of chemical reactions". Physica. 7 (4): 284–304. Bibcode:1940Phy.....7..284K. doi:10.1016/S0031-8914(40)90098-2. S2CID 33337019.

[2] Moyal, J. E. (1949). "Stochastic processes and statistical physics". Journal of the Royal Statistical Society. Series B (Methodological). 11 (2): 150–210. JSTOR 2984076.

[3] Gardiner, C. (2009). Stochastic Methods (4th ed.). Berlin: Springer. ISBN 978-3-642-08962-6.

[4] Van Kampen, N. G. (1992). Stochastic Processes in Physics and Chemistry. Elsevier. ISBN 0-444-89349-0.

[5] Risken, H. (1996). The Fokker–Planck Equation. Berlin, Heidelberg: Springer. pp. 63–95. ISBN 3-540-61530-X.

[6] 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.

[7] Pawula, R. F. (1967). "Generalizations and extensions of the Fokker–Planck–Kolmogorov equations" (PDF). IEEE Transactions on Information Theory. 13 (1): 33–41. doi:10.1109/TIT.1967.1053955.

[8] Pawula, R. F. (1967). "Approximation of the linear Boltzmann equation by the Fokker–Planck equation". Physical Review. 162 (1): 186–188. Bibcode:1967PhRv..162..186P. doi:10.1103/PhysRev.162.186.

[9] Risken, Hannes (6 December 2012). The Fokker-Planck Equation: Methods of Solution and Applications. ISBN 9783642968075.

[10] Rydin Gorjão, L.; Meirinhos, F. (2019). "kramersmoyal: Kramers--Moyal coefficients for stochastic processes". Journal of Open Source Software. 4 (44): 1693. arXiv:1912.09737. Bibcode:2019JOSS....4.1693G. doi:10.21105/joss.01693.

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@@ Line 23: / Line 23: @@
 Pawula theorem states that for any other choice of <math>m</math>, there exists a probability density function <math>\rho</math> that can become negative during its evolution <math>\partial_t\rho  \approx L_m \rho</math> (and thus fail to be a probability density function).<ref>{{cite journal |first=R. F. |last=Pawula |title=Generalizations and extensions of the Fokker–Planck–Kolmogorov equations |journal=IEEE Transactions on Information Theory |volume=13 |issue=1 |pages=33–41 |year=1967 |doi=10.1109/TIT.1967.1053955 |url=https://thesis.library.caltech.edu/8789/2/Pawula_rf_1965.pdf }}</ref><ref>{{cite journal |last=Pawula |first=R. F. |year=1967 |title=Approximation of the linear Boltzmann equation by the Fokker–Planck equation |journal=Physical Review |volume=162 |issue=1 |pages=186–188 |doi=10.1103/PhysRev.162.186 |bibcode=1967PhRv..162..186P }}</ref><ref>{{cite book|last1=Risken|first1=Hannes|title=The Fokker-Planck Equation: Methods of Solution and Applications|date=6 December 2012|isbn=9783642968075|url=https://books.google.com/books?id=dXvpCAAAQBAJ&q=Pawula-Theorem&pg=PA70}}</ref>
-However, this doesn't mean that Kramers-Moyal expansions truncated at for calculating distribution functions. Though the solution must have negative values at least for sufficiently small times, the resulting approximation probability density may still be better than the <math>m=2</math> approximation.
+However, this doesn't mean that Kramers-Moyal expansions truncated at other choices of <math>m</math> 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 <math>m=2</math> approximation.
 ==Implementations==