Is there a time constant or a characteristic time it takes for the diffusion equation
$$\frac{\partial \psi(\vec{x}, t)}{\partial t} = D \nabla^2 \psi(\vec{x},t)+V(\vec{x})$$
to reach steady-state conditions? You can assume the inhomogenous term $V(\vec{x})$ is time-independent, and that the problem is well-posed, and so on.