Consider the Cauchy problem given by
\begin{align}(1) \begin{cases} \partial_t u= -\partial_x [f(x)u],\;\; t\in [0,\|\Delta\|]\\ u(0,x)=\frac{1}{\sqrt{2\pi\|\Delta\|}}e^{-\frac{x^2}{2\|\Delta\|}}. \end{cases} \end{align}
I am trying to find out whether the latter is related in some way to the Fokker-Planck equation
\begin{align}(2) \begin{cases} \partial_t u= -\partial_x [f(x)u]+\frac{1}{2}\partial_{xx}u,\;\; t\in [0,\|\Delta\|]\\ u(0,x)=\delta(x), \end{cases} \end{align}
where $\|\Delta\|>0$.
If we let $f\equiv 0$ we have that the solution of $(1)$ is constant and equals the initial condition, on the other hand $(2)$ becomes the homogeneous heat equation, and its solution is given by the heat semigroup.
Then we have that for $t=\|\Delta\|$ the solution of $(1)$ and $(2)$ coincide.
I am wondering whether the latter holds also for the case in which $f$ is some well-behaved function, any idea?
EDIT:
Another example can be obtained by letting $f=1$, in that case $(1)$ becomes
\begin{align}(1) \begin{cases} \partial_t u= -\partial_x u,\;\; t\in [0,\|\Delta\|]\\ u(0,x)=\varphi(x)=\frac{1}{\sqrt{2\pi\|\Delta\|}}e^{-\frac{x^2}{2\|\Delta\|}}, \end{cases} \end{align} and the solution is given by $u(t,x)=\varphi(x-t)$ which at time $t=\|\Delta\|$ equals the density of a $N(\|\Delta\|,\|\Delta\|)$ r.v.
On the other hand $(2)$ becomes the Fokker-Planck equation of an Ito process given by $dX_t= dt+dB_t$ from where it's easy to see that at $t=\|\Delta\|$ again the solutions of $(1)$ and $(2)$ coincide.