# On the Fokker-Planck equation: deriving the transition PDF for small times

I report below (part of) page 73 of the book The Fokker-Planck Equation, by H. Risken

We now derive an expression for the transition probability density for small $$\tau$$ $$$$\tag{1} p(x,t+\tau|x', t)=\bigg(1+L_{\mathrm{FP}}(x,t)+O(\tau^2)\bigg)\delta(x-x')$$$$ with $$$$\tag{2} L_{\mathrm{FP}}(x,t):=-\frac{\partial}{\partial t}D_1(x,t)+\frac{\partial^2}{\partial x^2}D_2(x, t)$$$$ We get up to corrections of the order $$\tau^2$$: \begin{aligned} p(x,t+\tau|x', t)&=\bigg(1-\frac{\partial}{\partial t}D_1(x',t)\tau+\frac{\partial^2}{\partial x^2}D_2(x', t)\tau\bigg)\delta(x-x')\\ &=\exp\bigg(-\frac{\partial}{\partial t}D_1(x',t)\tau+\frac{\partial^2}{\partial x^2}D_2(x', t)\tau\bigg)\delta(x-x') \end{aligned} \tag{3}

I don't get the last equality in equation $$\mathrm{(3)}$$. At first glance, it would appear to be just a replacement. After all $$\exp(L_{\mathrm{FP}}\,\tau)\simeq 1+L_{\mathrm{FP}}\,\tau$$ for $$\tau\to 0$$. But is it?

After a few other steps, he writes the solution as

$$$$\tag{4} p(x,t+\tau|x',t)=\frac{1}{\sqrt{4\pi\,D_2(x',t)}}\exp\left(\frac{[(x-x')-D_1(x',t)\tau]^2}{4D_2(x', t)\tau}\right)$$$$

and then he says

for drift and diffusion coefficients independent of $$x$$ and $$t$$, $$\mathrm{(4)}$$ is not only valid for small $$\tau$$, but for arbitrary $$\tau>0$$ (the last line in equation $$\mathrm{(3)}$$ is then the formal solution).

So... did he just replace the approximate solution with the exact one? One is not supposed to know it ahead.

• Hi was my answer helpful at all? Feb 10 at 22:37

It is more than just a replacement of $$1+Lt\rightarrow \exp (Lt)$$. It is the following: within a region of $$x$$ and $$t$$ that $$D_{1,2}(x,t)$$ can be reliably approximated as a constant, the transition probability will be given by $$(4)$$. In other words, if we isolate our attention to such a confined region, we can reliably approximate the transition probability as $$(4)$$.
An analogous situation would be solving the differential equation $$y'(x) = \sin(x) y(x)$$ near the point $$x=\pi/2$$. The exact solution to this differential equation is $$y(x)=C e^{-\cos(x)}$$, and if we expand it about $$x=\pi/2$$ we get
$$y(x)\approx C e^{-\left(\frac{\pi}{2}-x\right)}$$
However in the vicinity of $$x=\pi/2$$ we can reliably approximate $$\sin (x)\approx 1$$, and therefore $$y(x)\approx C' e^{x}$$ near this point. Notice that from this approximate solution near $$x=\pi/2$$, you cannot deduce anything about $$y(x)$$ outside this region of validity.