actual hard math problem this time, relevant to something I'm trying to work on:

consider a situation where you have 1D diffusion with drift. The substance starts out with a distribution of p(x, t0) and evolves according to the Fokker-Planck equation:

d/dt [p(x,t)] = d^2/dx^2 [D(x)*p(x,t)] - d/dx [u(x)*p(x,t)]

visualization in a case with uniform diffusion coefficient and drift

the question is... does there exist some magical D(x) and u(x) functions so that, if p(x,t) is any Beta distribution, the future evolution will also always be a beta distribution?

In before someone says "set them both equal to zero"

beta distributions have compact support right?

@flawr they are only defined on the interval [0,1]

you could always rescale them

But I don't know if that matters because there's ways to force the density function to be zero at the edges by having an infinite drift strength or diffusion coefficient at the edges.

right but so you just consider the interval [0,1] as your domain?

can we allow to set D or u to zero?

I'd say so.

What exactly do you mean by rescaling? I think there's a way to map x -> x' such that either D(x') or u(x') is whatever function you want (like a constant).

you can rescale the distribution to another interval [a,b]

In my case, it's not physically meaningful for the "particles" to leave the boundary of [0,1] so rescaling over time could normally be a smart way to accomplish the goal but it doesn't work in my case.

Also it probably makes more sense to use an infinite drift u() rather than infinite diffusion D(x) to force the probability to be zero at the edges.