Partial Differential Equation $\frac{\partial}{\partial t} p(x,t) = \frac{\partial^2}{\partial x^2} \left[ x^2 p(x,t) \right]$

Question

In my research I have come across the partial differential equation \begin{equation} \frac{\partial}{\partial t} p(x,t) = \frac{\partial^2}{\partial x^2} \left[ x^2 p(x,t) \right]. \end{equation} Although I basically know how to solve such an equation, I would rather have a reference to a paper (due to limited space in journal). I assume that this must be a rather well-known equation with well-known solutions. Does this type of equation have a name? Does somebody know a paper where it is solved?

Answer 1

It's a 1-D Fokker-Planck equation with a Diffusion proportional to $x^2$ and $0$ drift term.

You can equally consider it as an imaginary time 1-D Schroedinger equation for a unit-charged particle coupled to an classical vector potential $A =\text{contstant} \times x$, choosing the Coulomb gauge. By the way, this problem has a well known analytical solution...