Kramers–Moyal expansion

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

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

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

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

where

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

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

Pawula theorem

The Pawula theorem states that the expansion either stops after the first term or the second term[6][7]. If the expansion continues past the second term it must contain an infinite number of terms, in order that the solution to the equation be interpretable as a probability density function.[8]

Implementations

• Implementation in Python: https://github.com/LRydin/KramersMoyal [9]

References

1. Kramers, H. A. (1940). "Brownian motion in a field of force and the diffusion model of chemical reactions". Physica. 7 (4): 284–304. doi:10.1016/S0031-8914(40)90098-2.
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). "Fokker–Planck equation". The Fokker–Planck Equation. Berlin, Heidelberg: Springer. pp. 63–95. ISBN 3-540-61530-X.
6. R. F. Pawula, "Generalizations and extensions of the Fokker- Planck-Kolmogorov equations," in IEEE Transactions on Information Theory, vol. 13, no. 1, pp. 33-41, January 1967, doi: 10.1109/TIT.1967.1053955.
7. Pawula, R. F. (1967). Approximation of the linear Boltzmann equation by the Fokker-Planck equation. Physical review, 162(1), 186.
8. Risken, Hannes. "The Fokker-Planck Equation: Methods of Solution and Applications".
9. Rydin Gorjão, L.; Meirinhos, F. (2019). "kramersmoyal: Kramers--Moyal coefficients for stochastic processes". Journal of Open Source Software. 4 (44): 1693. doi:10.21105/joss.01693.