Describing a boundary condition for Fokker-Planck equations I want to solve the following diffusion equation which is bounded on $(-1,1):$
$$ \partial_t P(x,t) = D\,\partial_{xx} P(x,t),$$
$$ P(x,0)=\delta(x-x_0)$$
where $D>0$ is the diffusion constant. I am assuming that once the process reaches $x=-1$ or $x=1$ it remains there forever. With this assumption, what boundary condition(s) should be imposed to the above PDE?
If $\tau=min\{t>0: |W_t|=1\}$ is the first time the standard Brownian motion reaches the boundary, then we know that $P(\tau < \infty)=1$. So it seems that we must impose some conditions like this:
$ \lim_{t \to \infty}P(-1,t|x_0,t_0)=\delta(-1)$ or $ \lim_{t \to \infty}P(1,t|x_0,t_0)=\delta(1)$.
But here in this paper, (and similar papers on the solutions of Fokker-Planck equations),  the absorbing boundary conditions on $(-1,1)$ are considered as
$P(-1,t|x_0,t_0) = P(1,t |x_0, t_0)=0$ for all $t$, which is not consistent with what I described above. How is this equivalent to the condition that I described?

Good question. A partial answer is that if you want to model the effect of particles "sticking" to the edges you have two choices: Either model by adding a "sticky" potential to the edges, or by assuming that the diffusion coefficient drops to zero at the edges. Usually the second case is preferred for technical reasons. In the second case, you can observe that this implies that $P=0,\partial_t P =0$ for $ |x|>1$ and the actual diffusion equation $\partial_t P =\partial_x (D(x)\partial_x P)$ has a singularity at the edges which implies $\partial_t P|_{-1,1}, \approx \partial_x P|_{-1,1}$ (that is, the flux is flowing through the domain edges and probability is being "lost") and this can be encoded in $P=0$ boundary conditions.
Adding a sticky potential is an interesting exercise, and I believe would yield a result that fits your intuition. I have seen such calculations done, and perhaps a more intrepid responder  can attempt this.
