Kramers–Moyal expansion explained

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] [3]} In many textbooks, the expansion is used only to derive the Fokker–Planck equation, and never used again. In general, continuous stochastic processes are essentially all Markovian, and so Fokker–Planck equations are sufficient for studying them. The higher-order Kramers–Moyal expansion only come into play when the process is jumpy. This usually means it is a Poisson-like process.^[4]

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 ^[5]

Statement

(\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|x _{0,} t _{0)} is the transition probability function, 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^{[6] [7] [8]} \partial _{t} p(x,t) = \sum ^{inf ty} _{n =1} ( - \partial ^{n[D} _{n(x,t)} p(x,t)]

and also$$\backslash partial\_t\; p(x,\; t|x\_0,\; t\_0)\; =\; \backslash sum\_^\backslash infty\; (-\backslash partial\_x)^n\; [D\_n(x,\; t)\; p(x,\; t|x\_0,\; t\_0)\; ]$$

where

D _{n(x,} t) are the Kramers–Moyal coefficients, defined by$$D\_n(x,\; t)\; =\; \backslash frac\backslash lim\_\; \backslash frac\; \backslash mu\_n(t|x,\; t-\backslash tau)$$and \mu _{n} are the central moment functions, defined by\mu _{n(t'} | x, t) = \int ^{inf ty} _{-inf ty} (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

D _{1} is the drift and D _{2} is the diffusion coefficient.^[9]

Also, the moments, assuming they exist, evolves as

$$\backslash frac\backslash left\backslash langle\; x^n\backslash right\backslash rangle=\backslash sum\_^n\; \backslash frac\backslash left\backslash langle\; x^\; D^(x,\; t)\backslash right\backslash rangle$$where angled brackets mean taking the expectation:

\left\langle f\right\rangle = \int f(x) p(x, t)dx .

n-dimensional version

The above version is the one-dimensional version. It generalizes to n-dimensions. (Section 4.7)

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):$$p(x)\; =\; \backslash frac\; \backslash int\; e^\backslash tilde\; p(k)dk\; =\; \backslash sum\_^\backslash infty\; \backslash frac\backslash delta^(x)\backslash mu\_n$$$$\backslash tilde\; p(k)\; =\; \backslash int\; e^\; p(x)\; dx\; =\; \backslash sum\_^\backslash infty\backslash frac\; \backslash mu\_n$$Similarly, $$p(x,\; t|\; x\_0,\; t\_0)\; =\; \backslash sum\_^\backslash infty\; \backslash frac\backslash delta^(x-x\_0)\; \backslash mu\_n(t|x\_0,\; t\_0)$$Now we need to integrate away the Dirac delta function. Fixing a small

\tau > 0 , we have by the Chapman-Kolmogorov equation,The n =0 term is just p(x, t - \tau) , so taking derivative with respect to time,$$\backslash partial\_t\; p(x,\; t)\; =\; \backslash lim\_\backslash frac\; 1\backslash tau\; \backslash sum\_^\backslash infty\; \backslash frac\; \backslash partial\_x^n\; (p(x,\; t-\backslash tau)\; \backslash mu\_n(t|x,\; t-\backslash tau))\; =\; \backslash sum\_^\backslash infty\; (-\backslash partial\_x)^n\; (p(x,\; t)\; D\_n(x,\; t))$$

The same computation with

p(x, t|x _{0,} t _{0)} gives the other equation.

Forward and backward equations

The equation can be recast into a linear operator form, using the idea of infinitesimal generator. Define the linear operator $$\backslash mathcal\; A\; f\; :=\; \backslash sum\_^\backslash infty\; (-\backslash partial\_x)^n[D\_n(x,t)\; f(x,t)]$$then the equation above states In this form, the equations are precisely in the form of a general Kolmogorov forward equation. The backward equation then states that$$\backslash partial\_t\; p(x\_1,\; t\_1|x,\; t)\; =\; -\backslash mathcal^\backslash dagger\; p(x\_1,\; t\_1|x,\; t)$$where$$\backslash mathcal\; A^\backslash dagger\; f\; :=\; \backslash sum\_^\backslash infty\; D\_n(x,t)\; \backslash partial\_x^n[f(x,t)]$$is the Hermitian adjoint of

l A .

Computing the Kramers–Moyal coefficients

By definition,$$D\_n(x,\; t)\; =\; \backslash frac\backslash lim\_\; \backslash frac\; \backslash mu\_n(t|x,\; t-\backslash tau)$$This definition works because

\mu _{n(t|x,} t) = 0 , as those are the central moments of the Dirac delta function. Since the even central moments are nonnegative, we have D _{2n} \geq 0 for all n\geq 1 . When the stochastic process is the Markov process dX = bdt + \sigma dW _{t} , we can directly solve for p(x, t|x, t - \tau) as approximated by a normal distribution with mean x + b(x)\tau and variance \sigma ^{2\tau} . This then allows us to compute the central moments, and so$$D\_1\; =\; b,\; \backslash quad\; D\_2\; =\; \backslash frac\; 12\; \backslash sigma^2,\; \backslash quad\; D\_3=D\_4=\backslash cdots\; =\; 0$$This then gives us the 1-dimensional Fokker–Planck equation:$$\backslash partial\_t\; p\; =\; -\backslash partial\_x(bp)\; +\; \backslash frac\; 12\; \backslash partial\_x^2(\backslash sigma^2\; p)$$

Pawula theorem

Pawula theorem states that either the sequence

D _{1,} D _{2,} D _{3,} ... becomes zero at the third term, or all its even terms are positive.^{[10] [11]}

Proof

By Cauchy–Schwarz inequality, the central moment functions satisfy

\mu ^{2} _{n + m} \leq \mu _{2n} \mu _{2m} . So, taking the limit, we have D ^{2} _{n + m} \leq (( 2 n)!( 2 m)!)/((n + m)! ^{2}) D _{2n} D _{2m} . If some D _{2+n} ≠ 0 for some n \geq 1 , then D _{2} D _{2+2n} > 0 . In particular, D _{2+n} , D _{2+2n} , D _{2+4n} , ... > 0 . So the existence of any nonzero coefficient of order \geq 3 implies the existence of nonzero coefficients of arbitrarily large order. Also, if D _{n} ≠ 0 , then D _{2D} _{2n-2} > 0 , D _{4D} _{2n-4} > 0 , ... . So the existence of any nonzero coefficient of order n implies all coefficients of order 2 , 4 , ..., 2 n -2 are positive.

Interpretation

Let the operator

l A _{m} be defined such l A _{m} f : = \sum ^{m} _{n =1} ( - \partial ^{n[D} _{n(x,t)} f(x,t)] . The probability density evolves by \partial _{t\rho} ≈ l A _{m} \rho . Different order of m gives different level of approximation.m = 0 : the probability density does not evolvem =1 : it evolves by deterministic drift only.m =2 : it evolves by drift and Brownian motion (Fokker-Planck equation)m =inf ty : the fully exact equation.

Pawula theorem means that if truncating to the second term is not exact, that is,

l A _{2} ≠ l A , then truncating to any term is still not exact. Usually, this means that for any truncation l A _{m} , there exists a probability density function \rho that can become negative during its evolution \partial _{t\rho} ≈ l A _{m} \rho (and thus fail to be a probability density function). 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.

Notes and 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 . 10.1016/S0031-8914(40)90098-2 . 1940Phy.....7..284K . 33337019 .
2. Moyal . J. E. . 1949 . Stochastic processes and statistical physics . . Series B (Methodological) . 11 . 2 . 150–210 . 10.1111/j.2517-6161.1949.tb00030.x . 2984076 .
3. Book: Risken . Hannes . The Fokker-Planck Equation: Methods of Solution and Applications . 6 December 2012 . Springer . 9783642968075.
4. Book: Spinney . Richard . Ford . Ian . Klages . Rainer . Just . Wolfram . Jarzynski . Christopher . 1201.6381 . Fluctuation relations: a pedagogical overview . 10.1002/9783527658701.ch1 . 978-3-527-41094-1 . Weinheim . 3308060 . 3–56 . Wiley-VCH . Reviews of Nonlinear Dynamics and Complexity . Nonequilibrium Statistical Physics of Small Systems: Fluctuation relations and beyond . 2013.
5. Rydin Gorjão . L. . Meirinhos . F. . 2019 . kramersmoyal: Kramers--Moyal coefficients for stochastic processes . . 4 . 44 . 1693 . 1912.09737 . 2019JOSS....4.1693G . 10.21105/joss.01693 . free.
6. Book: Gardiner, C. . 2009 . Stochastic Methods . 4th . Berlin . Springer . 978-3-642-08962-6 .
7. Book: Van Kampen, N. G. . 1992 . Stochastic Processes in Physics and Chemistry . Elsevier . 0-444-89349-0 .
8. Book: Risken, H. . 1996 . The Fokker–Planck Equation . 63–95 . Springer . Berlin, Heidelberg . 3-540-61530-X .
9. Book: Wolfgang . Paul . Jörg . Baschnagel . A Brief Survey of the Mathematics of Probability Theory . Stochastic Processes . 17–61 [esp. 33–35] . Springer . 2013 . 978-3-319-00326-9. 10.1007/978-3-319-00327-6_2 .
10. Pawula . R. F. . 1967 . Generalizations and extensions of the Fokker–Planck–Kolmogorov equations . IEEE Transactions on Information Theory . 13 . 1 . 33–41 . 10.1109/TIT.1967.1053955.
11. Pawula . R. F. . 1967 . Approximation of the linear Boltzmann equation by the Fokker–Planck equation . Physical Review . 162 . 1 . 186–188 . 1967PhRv..162..186P . 10.1103/PhysRev.162.186.