Relating Fokker-Planck to stochastic differential equations

(Adapted from Risken §3.3.2)

Recall that the FPE coefficients \(D^{(1)}\) and \(D^{(2)}\) are directly related to the moments of \(p(t+Δt, x' \mid t, x)\):

\(D^{(n)} = \frac{1}{n! Δt} \langle ΔX(t)^n\rangle\)

(7)\[\begin{split}\begin{aligned} \frac{1}{n!} M_n(t,Δt,n) &= \frac{1}{n!} \langle \bigl(X(t+Δt) - X(t)\bigr)^n\rangle \\ &= \frac{1}{n!} \langle ΔX(t)^n\rangle \\ &= D^{(n)}(t, x)Δt + \mathcal{O}(Δt^2) \,. \end{aligned}\end{split}\]

We want to relate these to the drift and diffusion terms of the Langevin equation:

\[dX(t) = f(t,X(t)) dt + g(t,X(t)) dW \,.\]

Recall that depending the chosen convention, integrating this expression over a small interval \(Δt\) produces different results:

Itô	\(ΔX(t_i) = f(t_i,X(t_i))Δt + g(t_i,X(t_i))ΔW(t_i)\hphantom{+ g(t_{i+1},X(t_{i+1}))}\)
Stratonovich	\(\begin{aligned} ΔX(t_i) &= f(t_i,X(t_i))Δt + \frac{g(t_i,X(t_i)) + g(t_{i+1},X(t_{i+1}))}{2}ΔW(t_i) \\ &= f(t_i,X(t_i))Δt + g(t_i,X(t_i))ΔW(t_i) + \frac{1}{2} \left.\frac{\partial g}{\partial x}\right\rvert_{t_i}ΔX(t_i) ΔW(t_i) \\ &= f(t_i,X(t_i))Δt + g(t_i,X(t_i))ΔW(t_i) + \frac{1}{2} \left.\frac{\partial g}{\partial x}\right\rvert_{t_i}g(t_i,X(t_i)) ΔW(t_i)^2 + \mathcal{O}(Δt^{3/2}) \end{aligned}\)
Hänggi	\(\begin{aligned} ΔX(t_i) &= f(t_i,X(t_i))Δt + g(t_{i+1},X(t_{i+1}))ΔW(t_i)\hphantom{+ g(t_{i},X)} \\ &= f(t_i,X(t_i))Δt + g(t_i,X(t_i))ΔW(t_i) + \hphantom{\frac{1}{2}} \left.\frac{\partial g}{\partial x}\right\rvert_{t_i}g(t_i,X(t_i)) ΔW(t_i)^2 + \mathcal{O}(Δt^{3/2}) \end{aligned}\)

Caution

Risken uses the convention \(\langle ΔW(t) ΔW(t')\rangle = 2\, δ(t)\, Δt\,.\)

We also have the following rules for stochastic calculus, which are valid in any convention:

\[\begin{split}\begin{aligned} \langle ΔW(t) Δt \rangle = \langle Δt^2 \rangle &= 0 \\ \langle ΔW(t) \rangle &= 0 \\ \langle ΔW(t) ΔW(t') \rangle &= δ(t)\, Δt \\ \langle g(t, X(t)) ΔW(t) \rangle &= g(t, X(t)) \langle ΔW(t) \rangle \,. \end{aligned}\end{split}\]

Applying these rules we can evaluate the expectations in Eq. (7).

Itô convention

\[\begin{split}\begin{aligned} \langle ΔX(t_i) \rangle &= f(t_i) Δt + g(t_i, X(t_i)) \underbrace{\langle ΔW(t_i) \rangle}_{=0} \\ &= f(t_i) Δt \\ \langle ΔX(t_i)^2 \rangle &= g(t_i, X(t_i))^2 \underbrace{\langle ΔW(t_i)^2 \rangle}_{=Δt} + \mathcal{O}(ΔtΔW(t_i)) \\ &= g(t_i, X(t_i))^2 Δt \end{aligned}\end{split}\]

Stratonovich convention

\[\begin{split}\begin{aligned} \langle ΔX(t_i) \rangle &= f(t_i) Δt + g(t_i, X(t_i)) \underbrace{\Bigl\langle ΔW(t_i) \Bigr\rangle}_{=0} + \frac{1}{2} \frac{\partial g(t_i, X(t_i))}{\partial x} g(t_i, X(t_i)) \underbrace{\Bigl\langle ΔW(t_i)^2 \Bigr\rangle}_{=Δt} \\ &= f(t_i) Δt + \frac{1}{2} \frac{\partial g(t_i, X(t_i))}{\partial x} g(t_i, X(t_i)) Δt \\ \langle ΔX(t_i)^2 \rangle &= g(t_i, X(t_i))^2 Δt \end{aligned}\end{split}\]

Hänggi-Klimontovich convention

\[\begin{split}\begin{aligned} \langle ΔX(t_i) \rangle &= f(t_i) Δt + \hphantom{\frac{1}{2}} \frac{\partial g(t_i, X(t_i))}{\partial x} g(t_i, X(t_i)) Δt \\ \langle ΔX(t_i)^2 \rangle &= g(t_i, X(t_i))^2 Δt \end{aligned}\end{split}\]

Substituting these results back into Eq. (7), we get the correspondences listed in Table 1.

Caution

If the convention \(\langle dW(t) dW(t')\rangle = 2\, δ(t)\, dt\) is used, the spurious drift terms must be multiplied by 2.

Table 1 Correspondence between Fokker-Planck and Langevin coefficients.

	drift, \(D^{(1)}\)	diffusion, \(D^{(2)}\)
Itô	\(f(t_i, X(t_i))\)	\(\frac{1}{2} \,g(t_i, X(t_i))^2\)
Stratonovich	\(f(t_i, X(t_i)) + \frac{1}{2} \frac{\partial g(t_i, X(t_i))}{\partial x} g(t_i, X(t_i))\)	\(\frac{1}{2} \,g(t_i, X(t_i))^2\)
Hänggi-Klimontovich	\(f(t_i, X(t_i)) + \hphantom{\frac{1}{2}} \frac{\partial g(t_i, X(t_i))}{\partial x} g(t_i, X(t_i))\)	\(\frac{1}{2} \,g(t_i, X(t_i))^2\)

The additional drift appearing in the expressions under the Stratonovich and Hänggi-Klimontovich conventions is sometimes called the spurious drift, or the noise-induced drift. It only comes into play when \(g\) depends directly on \(X\).