Consider a simple 1D particle diffusion process described by the SDE $dx=\sigma dW$, where $dW$ is a Wiener process. The forward Fokker-Planck equation can then be written as $$ \frac{\partial P(x,t)}{\partial t} = D \frac{\partial^2 P(x,t)}{dx^2}, $$ where $D=\sigma^2/2$.
There are two boundaries at $x=0$ and $x=L$ respectively. At $x=0$, there is a source that generates particles with the same initial condition $x(0)=0$ and $\dot x(0)=1$. $x=0$ is an absorbing boundary at the same time, meaning that if any particle hits $x=0$ except when it was generated there, it will be absorbed and disappear from the system. $x=L$ is also an absorbing boundary with similar behaviour, except that there is no source located there. Assuming that the particles are generated at the source at a constant, fast rate (compared to the typical time of a particle remaining in the system). What is the steady-state density of the particles in the system?
The problem is how should the boundary condition at $x=0$ be set up. At an absorbing boundary, the probability is zero, but it is obviously not so in this case due to the existence of the source. If we set up the boundary condition with the help of a probability current $\vec{J}$, which comes from the Fokker-Planck equation by rewriting it into $\partial_t P(x,t) + \operatorname{div} \vec{J}=0$, how should it be done? Will it be correct if we simply let that $J(x=0)=\alpha$ for some constant $\alpha$ at the steady-state and establish the fact that $J(x)=\alpha$ for all $0 \leq x \leq L$? I obtained the steady-state probability density $P_s(x)$ in this form using this approach: $$ P_s(x)=\frac{2}{L}-\frac{2x}{L^2}. $$
But is this boundary condition resonable?
And if we extend the system to a 2D rotational diffusion (we omit $y$ because it is not coupled with the other variables and the probability density should be constant along $y$) system, where the Ito equations look like the following: $$ dx = v \cos \theta\; dt \\ d\theta = \sigma \; dW, $$ and the walls at $x=0$ and $x=L$ are infinitely long (along $y$-direction) absorbing boundaries, while the wall $x=0$ is a source constantly emits particles (all initially with $\theta(0)$=0) at all values of $y$. The Fokker-Planck equation obtained is $$ \frac{\partial P(x,\theta,t)}{\partial t} = -v \cos \theta \frac{\partial P(x,\theta,t)}{\partial x} + D \frac{\partial^2 P(x,\theta,t)}{\partial \theta^2}, $$ where $D=\sigma^2/2$. How the boundary conditions for $x$ and $\theta$ should be set up this time?