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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]

Start with the integro-differential master equation

{\frac {\partial p(x,t)}{\partial t}}=\int p(x,t|x_{0},t_{0})p(x_{0},t_{0})dx_{0}

where $p(x',t'\mid x,t)$ is the transition probability, and $p(x,t)$ is the probability density at time $t$

The Kramers–Moyal expansion transforms the above to an infinite order partial differential equation^[3]^[4]^[5]

\partial _{t}p(x,t|x',t')=\sum _{n=1}^{\infty }{\frac {(-1)^{n}}{n!}}{\frac {\partial ^{n}}{\partial x^{n}}}[\mu _{n}(t'|x,t)p(x,t)]

where $\mu _{n}$ is the central moment function defined by

\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 $\alpha _{1}$ is the drift and $\alpha _{2}$ is the diffusion coefficient.^[6]

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): $p(x)={\frac {1}{2\pi }}\int e^{-ikx}{\tilde {p}}(k)dk=\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{n!}}\delta ^{(n)}(x)\mu _{n}$ ${\tilde {p}}(k)=\int e^{ikx}p(x)dx={\frac {(ik)^{n}}{n!}}\mu _{n}$

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 1: / Line 1: @@
-In [[stochastic processes]], '''Kramers–Moyal expansion''' refers to a [[Taylor series]] expansion of the [[master equation]], named after [[Hans Kramers]] and [[José Enrique Moyal]].<ref>{{cite journal |last=Kramers |first=H. A. |year=1940 |title=Brownian motion in a field of force and the diffusion model of chemical reactions |journal=Physica |volume=7 |issue=4 |pages=284–304 |doi=10.1016/S0031-8914(40)90098-2 |bibcode=1940Phy.....7..284K |s2cid=33337019 }}</ref><ref>{{cite journal |last=Moyal |first=J. E. |year=1949 |title=Stochastic processes and statistical physics |journal=[[Journal of the Royal Statistical Society]] |series=Series B (Methodological) |volume=11 |issue=2 |pages=150–210 |jstor=2984076 }}</ref> This expansion transforms the [[integro-differential equation|integro-differential]] [[master equation]]
+In [[stochastic processes]], '''Kramers–Moyal expansion''' refers to a [[Taylor series]] expansion of the [[master equation]], named after [[Hans Kramers]] and [[José Enrique Moyal]].<ref>{{cite journal |last=Kramers |first=H. A. |year=1940 |title=Brownian motion in a field of force and the diffusion model of chemical reactions |journal=Physica |volume=7 |issue=4 |pages=284–304 |doi=10.1016/S0031-8914(40)90098-2 |bibcode=1940Phy.....7..284K |s2cid=33337019 }}</ref><ref>{{cite journal |last=Moyal |first=J. E. |year=1949 |title=Stochastic processes and statistical physics |journal=[[Journal of the Royal Statistical Society]] |series=Series B (Methodological) |volume=11 |issue=2 |pages=150–210 |jstor=2984076 }}</ref>
+Start with the [[integro-differential equation|integro-differential]] [[master equation]]
-:<math>\frac{\partial p(x,t)}{\partial t} =\int dx'[W(x|x')p(x',t)-W(x'|x)p(x,t)]</math>
+:<math>\frac{\partial p(x,t)}{\partial t} =\int p(x,t|x_0, t_0)p(x_0, t_0) dx_0</math>
-where <math>p(x,t|x_0,t_0)</math> (for brevity, this probability is denoted by <math>p(x,t)</math>) is the transition probability density, to an infinite order [[partial differential equation]]<ref>{{cite book |last=Gardiner |first=C. |year=2009 |title=Stochastic Methods |edition=4th |location=Berlin |publisher=Springer |isbn=978-3-642-08962-6 }}</ref><ref>{{cite book |last=Van Kampen |first=N. G. |year=1992 |title=Stochastic Processes in Physics and Chemistry |publisher=Elsevier |isbn=0-444-89349-0 }}</ref><ref>{{cite book |last=Risken |first=H. |year=1996 |title=The Fokker–Planck Equation |pages=63–95 |publisher=Springer |location=Berlin, Heidelberg |isbn=3-540-61530-X }}</ref>
+where <math>p(x', t'\mid x, t)</math> is the [[Transition rate matrix|transition probability]], and <math>p(x,t)</math> is the probability density at time <math>t</math>
-:<math>\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)]</math>
+The Kramers–Moyal expansion transforms the above to an infinite order [[partial differential equation]]<ref>{{cite book |last=Gardiner |first=C. |year=2009 |title=Stochastic Methods |edition=4th |location=Berlin |publisher=Springer |isbn=978-3-642-08962-6 }}</ref><ref>{{cite book |last=Van Kampen |first=N. G. |year=1992 |title=Stochastic Processes in Physics and Chemistry |publisher=Elsevier |isbn=0-444-89349-0 }}</ref><ref>{{cite book |last=Risken |first=H. |year=1996 |title=The Fokker–Planck Equation |pages=63–95 |publisher=Springer |location=Berlin, Heidelberg |isbn=3-540-61530-X }}</ref>
-where
-:<math>\alpha_n(x) = \int_{-\infty}^\infty (x'-x)^n W(x'\mid x) \ dx'.</math>
+:<math>\partial_t p(x,t|x', t') = \sum_{n=1}^\infty \frac{(-1)^n}{n!} \frac{\partial^n}{\partial x^n}[\mu_n(t' | x, t) p(x,t)]</math>
+where <math>\mu_n</math> is the [[central moment]] function defined by
-Here <math>W(x'\mid x)</math> is the [[Transition rate matrix|transition probability rate]]. The [[Fokker–Planck equation]] is obtained by keeping only the first two terms of the series in which <math>\alpha_1</math> is the [[drift velocity|drift]] and <math>\alpha_2</math> is the diffusion coefficient.<ref>{{cite book |first=Wolfgang  |last=Paul |first2=Jörg |last2=Baschnagel |chapter=A Brief Survey of the Mathematics of Probability Theory |title=Stochastic Processes |pages=17–61 [esp. 33–35] |publisher=Springer |year=2013 |isbn= |doi=10.1007/978-3-319-00327-6_2 }}</ref>
+:<math>\mu_n(t' | x, t) = \int_{-\infty}^\infty (x'-x)^n p(x', t'\mid x, t) \ dx'.</math>
+The [[Fokker–Planck equation]] is obtained by keeping only the first two terms of the series in which <math>\alpha_1</math> is the [[drift velocity|drift]] and <math>\alpha_2</math> is the diffusion coefficient.<ref>{{cite book |first=Wolfgang  |last=Paul |first2=Jörg |last2=Baschnagel |chapter=A Brief Survey of the Mathematics of Probability Theory |title=Stochastic Processes |pages=17–61 [esp. 33–35] |publisher=Springer |year=2013 |isbn= |doi=10.1007/978-3-319-00327-6_2 }}</ref>
+== 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 (probability theory)|characteristic function]] page):<math display="block">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
+</math><math display="block">
+\tilde p(k) = \int e^{ikx} p(x) dx =  \frac{(ik)^n}{n!} \mu_n </math>
 ==Pawula theorem==