I have quickly done some basic mathematics that can probably be used by friends to solve the fokker planck PDEs. First we are dealing with densities of SDEs where the density vanishes at zero. We will improve that later to include the case of densities that reach zero. We suppose that we choose a certain point on the density of SDE and this point evolves in times so that the CDF of the SDE density remains constant. Let us suppose that we denote the density of SDE as [$]p(x,t)[$] and the CDF as [$]P(x,t)[$] and we denote the time changing constant CDF points as [$]v(t)[$]. We have the following integrals

[$]\int_0^{v(t)} p(x,t) dx =P(v,t)[$]

We also know that this point preserves the CDF so that

[$]\int_0^{v(t)} \frac{\partial p(x,t)}{\partial t} dx=0[$]

we write a general Fokker planck equation in Lamperti coordinates as

[$]-\frac{\partial [\mu(x) p(x,t)]}{\partial x} + .5 {\sigma}^2 \frac{\partial ^2 p(x,t)}{\partial x^2} = \frac{\partial p(x,t)}{\partial t}[$]

we integrate both sides of the partial differential equation from 0 to [$]v(t)[$] which is the constant CDF point(or constant CDF curve in time)

[$]- \int_0^{v(t)} \frac{\partial [\mu(x) p(x,t)]}{\partial x} dx+ .5 {\sigma}^2 \int_0^{v(t)} \frac{\partial ^2 p(x,t)}{\partial x^2} dx= \int_0^{v(t)} \frac{\partial p(x,t)}{\partial t}dx[$]

but [$] \int_0^{v(t)} \frac{\partial p(x,t)}{\partial t}dx= \frac {\partial P(v,t)}{\partial t}=0 [$] so we are left with the left hand side of the equation and the right hand side goes to zero and

[$]- \int_0^{v(t)} \frac{\partial [\mu(x) p(x,t)]}{\partial x} dx+ .5 {\sigma}^2 \int_0^{v(t)} \frac{\partial ^2 p(x,t)}{\partial x^2} dx= 0[$]

differentiating both sides by t, we get by Leibniz integral rule

[$]- \frac{\partial [\mu(v) p(v,t)]}{\partial v} \frac{\partial v}{\partial t}- \int_0^{v(t)} \frac{\partial^2 [\mu(x) p(x,t)]}{\partial x \partial t} dx[$]

[$]+ .5 {\sigma}^2 \frac{\partial ^2 p(v,t)}{\partial v^2} \frac{\partial v}{\partial t} + .5 {\sigma}^2 \int_0^{v(t)} \frac{\partial ^3 p(x,t)}{\partial x^2 \partial t} dx=0[$]

It is the [$]\frac{\partial v}{\partial t}[$] that tells us about the evolution of constant CDF points. I believe we can we very easily and recursively do these integrations going from one constant CDF point to the next constant CDF point and determining their evolution one by one. By construction, we know the CDF on each of the constant CDF points and its derivatives can be found by numerical differentiation and the partial differential with respect to t can be calculated by recursively using the original PDE. I will come back with more experiments in a few days.