I'm learning about boundary conditions for the Fokker-Planck equation and am misunderstanding something fundamental.
In one dimension, the Fokker-Planck equation is $$\partial_t f(x,t) = - \partial_x j(x,t)$$ where $f(x,t)$ is the distribution of $x$ and $j(x,t)$ is the probability flux: $$j(x,t) = A(x,t)f(x,t) - \partial_x B(x,t)f(x,t)$$ My textbook (Gardiner) says that the boundary condition for a reflecting barrier at $x=a$ is $$0 = j(a,t)$$
My question is: how can we reconcile this boundary condition with arbitrary initial distributions? I.e. there could be many initial distributions $f(x,0)$ such that $j(x,0)\neq 0$. Is the answer that the boundary conditions only hold for $t>0$?
As an example: consider the case with reflecting barriers at $a$ and $b$, and $x$ is a Wiener process with no drift, so that $A(x,t)=0$ and $B(x,t)=B$. In this case the flux is: $$j(x,t) = - B \partial_x f(x,t)$$ and the Fokker-Planck equation is $$\partial_t f(x,t) = B \partial_x^2 f(x,t)$$ The boundary conditions for the reflecting barriers are $$ 0 = B \partial_x f(a,t) = B \partial_x f(b,t)$$
The stationary distribution on $[a,b]$ is clearly $f(x)=\frac{1}{b-a}$. But suppose the initial condition is something non-constant but still smooth, e.g. $f(x,0)=c e^{-x}$ (for an appropriate integrating coefficient $c$). In this case, the flux is $$j(x,0) = B f(x,0)$$ which is non-zero at the barriers $a$ and $b$, even though they must be reflecting.
Thanks!
Just considering the PDE, what will happen is that a discontinuity in the derivatives will occur at $(x,t) = (a,0)$, which will be evolved forward by the flow. For the diffusive case, this will smooth out, but for a purely drift case, the discontinuity will remain.
What is happening here is that often the PDE can be defined in a more general way (weak form) that allows for unique solutions when the initial conditions are not smooth, or not even continuous is some cases. In this setting, the boundary conditions are only applied for $t>0$. If the initial conditions match the boundary conditions, then the limit $\lim_{t\to0} f(\cdot,t) = f(0,\cdot)$ will hold in stronger topologies, e.g., smooth or continuous functions. However, this is not necessary for the evolution equation to be well-defined, although it requires some technical mathematics to get to that point.