Is it possible for the solution of the Fokker-Planck equation to have negative values? I am referring to the mathematical aspect, irrespective of its physical interpretation. Additionally, considering that the solution represents a probability distribution function, is it acceptable to impose a constraint ensuring that the solution remains strictly positive?
Hi, Faezeh.
I feel like I don't fully understand your question. When I have faced non-correct Fokker-Planck solutions, such as regions with very high fluctuations and solutions close to zero but negative at the extremes, I used a smaller temporal and spatial step plus an order 8 Runge-Kutta integration scheme This solved the problems. So I assume that you are observing numerical instabilities due to the non-linearity of the problem you are dealing with.
https://en.wikipedia.org/wiki/Fokker–Planck_equation
The answer depends on exactly what you are prepared to count as being a "Fokker-Planck equation" when extracted from its original "physical interpretation". In a very abstracted form of the equation, if "p" is a solution, them "-p" is also a solution, given a corresponding change to the initial condition. (Similarly going from "p" to "Kp" for any K.) One could also start from initial conditions that are not a probability distribution. Other versions might allow the diffusion coefficient term to be negative.
As to your question "is it acceptable to impose a constraint ensuring that the solution remains strictly positive", the seemingly standard situation of having fixed boundaries provides an example where the solution is exactly zero at and outside those boundaries. Presumably one can still say that the "Fokker-Planck equation" applies outside those boundaries. Perhaps this "fixed boundary" case provides a good example of where you need to supply all the conditions for the problem rather than just the differential equation part.
So it seems that the probability distribution part of the problem is rather intrinsic to specifying the complete problem. In the simplest cases it looks as if working in terms of characteristic functions is a route to formal solutions, and this may or may not help with non-simple cases.
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