I stumbled upon a continuity equation with a $\nabla^2$ term on the right-hand side:
$$ \partial_t \rho + \nabla (\vec b \rho) = D \nabla^2 \rho , $$
where $b$ denotes the forward velocity and $D$ is a constant.
What's the meaning of such a diffusion equation?
Since, we have particle number conservation, we have
$$ \partial_t \rho + \nabla (\vec v \rho) = 0 , $$
where $v$ denotes the ordinary flux velocity. Moreover, if there are sources, we have
$$ \partial_t \rho + \nabla (\vec v \rho) = \sigma . $$