Derivation of Fokker-Planck equation (Kramers-Moyal expansion)#

(This page is heavily based on Sections 4.1, 4.3 and 4.7 in Risken.)

Kramers-Moyal expansion (1D)#

(2)#\[\begin{split}\begin{aligned} p(t + τ, x) &= \int p(t + τ, x | x', t) p(t, x') dx' \\ &\approx p(t, x) + \frac{\partial p(t, x)}{\partial t} τ + \mathcal{O}(τ^2) \,. \end{aligned}\end{split}\]

Goal

Obtain expression for \(p(t + τ, x | x', t)\) for small \(τ\).

Assuming all moments of \(p(t + τ, x | x', t)\) exist, we can formally write them

\[M_n(t,τ,x') := \int (x-x')^n p(τ+τ, x | t, x') dx \,.\]

We now construct the characteristic function

\[\begin{split}\begin{aligned} \varphi(t,τ, u,x') &= \int_{-\infty}^{\infty} e^{iu(x-x')} p(t+τ,x \mid t, x') dx \\ &= 1 + \sum_{n=1}^{\infty} \frac{(iu)^n}{n!} M_n(t,τ, x') \,; \end{aligned}\end{split}\]

and apply the inverse transform to get back the probability

(3)#\[\begin{split}\begin{aligned} p(t+τ,x | t, x') &= \frac{1}{2π} \int_{-\infty}^\infty e^{-iu(x-x')} \varphi(t,τ,u,x') du \\ &= \frac{1}{2π} \int_{-\infty}^\infty e^{-iu(x-x')} \left[1 + \sum_{n=1}^{\infty} \frac{(iu)^n}{n!} M_n(t,τ, x') \right] du \,. \end{aligned}\end{split}\]

With the following two identities,

Identities

\[\begin{split}\begin{aligned} \frac{1}{2π} \int_{-\infty}^\infty (iu)^n e^{-iu(x-x')} du &= \left( -\frac{\partial}{\partial x} \right)^n δ(x - x') \\[1.5ex] δ(x-x')f(x') &= δ(x-x')f(x) \end{aligned} \qquad\text{(for $n \geq 0$)} \end{split}\]

we can rewrite (3) as

(4)#\[ p(t+τ,x | t, x') = \left[1 + \sum_{n=1}^{\infty} \left(\frac{\partial}{\partial x}\right)^n M_n(t,τ, x') \right] δ(x-x') \,. \]

Substituting back into (2),

\[\begin{split}\begin{aligned} p(t + τ, x) - p(t, x) &= \frac{\partial p(t, x)}{\partial t} τ + \mathcal{O}(τ^2) \\ \int p(t + τ, x | x', t) p(t, x') dx' - p(t, x) &= p(t, x) - p(t, x) + \sum_{n=1}^{\infty} \left(\frac{\partial}{\partial x}\right)^n M_n(t,τ, x) p(t, x) \,. \end{aligned}\end{split}\]

We now assume that each moment can itself be expanded to first order:

\[\frac{1}{n!} M_n(t,τ,n) = D^{(n)}(t, x)τ + \mathcal{O}(τ^2) \,,\]

Note

The zero order term must vanish, since we have \(p(t,x|t,x') = δ(x-x')\) and therefore all moments vanish at \(t=t'\).

which leads to the Kramers-Moyal expansion:

(5)#\[\frac{\partial p(t,x)}{\partial t} = \sum_{n=1}^\infty \left(-\frac{\partial}{\partial x}\right)^n D^{(n)}(t,x) p(t,x)\,.\]

Pawula theorem#

The Kramers-Moyal expansion (Eq. (5)) either

  • stops on the first term;

  • stops on the second term;

  • contains an infinite number of terms.

If it stops after the second term it is called a Fokker-Planck equation.

A drift-diffusion process with Gaussian noise has an associated Fokker-Planck equation.

A jump process does not have an associated Fokker-Planck equation.

Fokker-Planck in higher dimensions#

(6)#\[ \frac{\partial p(t, x)}{\partial t} = \underbrace{- \sum_i \frac{\partial}{\partial x_i} \left[D_i^{(1)}(x) \, p(t,x)\right]}_{\text{drift}} + \underbrace{\sum_{i,j} \frac{\partial^2}{\partial x_i \partial x_j} \left[D_{ij}^{(2)}(x) \, p(t, x) \right]}_{\text{diffusion}} \]