In my current work on stochastic dynamics, I have stumbled upon the following problem, which I will present in a simplified form. I have two (or more) different Fokker-Planck equations, all of which I can solve individually. The system should switch stochastically (in simple cases with a unrelated Poisson-process) between the different FP-equations.
Here is a simple example case:
Process A: $$\frac{\partial}{\partial t} P(x,t) = - A\frac{\partial}{\partial x}P(x,t)+D_A\frac{\partial^2}{\partial x^2} P(x,t) $$
Process B: $$\frac{\partial}{\partial t} P(x,t) = - B\frac{\partial}{\partial x}P(x,t)+D_B\frac{\partial^2}{\partial x^2} P(x,t) $$
Where $A,B$ are real $D_A$ and $D_B$ real positive constants and the switching happens according to an unrelated poisson process with rates: $$q_{on} \text{: A --> B}$$ $$q_{off} \text{: B --> A}$$
In the future I would like to include dependence of the switching process on the system state, i.e. x. Is there a framework to study these types of processes, is there literature on it? I am a physics undergrad.