So the probability density of Brownian motion on $\mathbb{R}$ is described by the heat equation on $\mathbb{R}$. So the Brownian motion is the SDE corresponding to the heat equation PDE.
How do we know what is the SDE corresponding to the heat equation on $[0,1]$ with Dirichlet or Neumann boundary conditions? I saw somewhere that that the Neumann BCs might correspond to a reflecting boundary for the SDE? Is this true for other PDEs (say, Fokker-Planck or reaction-diffusion equation)?