  The Laplacian $\Delta$ is the infinitesimal generator of isotropic Brownian
motion, being the limit process of normal diffusion, while the fractional
Laplacian $\Delta^{\beta/2}$ serves as the infinitesimal generator of the limit
process of isotropic L\'{e}vy process. Taking limit, in some sense, means that
the operators can approximate the physical process well after sufficient long
time. We introduce the nonlocal operators (being effective from the starting
time), which describe the general processes undergoing normal diffusion. For
anomalous diffusion, we extend to the anisotropic fractional Laplacian
$\Delta_m^{\beta/2}$ and the tempered one $\Delta_m^{\beta/2,\lambda}$ in
$\mathbb{R}^n$. Their definitions are proved to be equivalent to an alternative
one in Fourier space. Based on these new nonlocal diffusion operators, we
further derive the deterministic governing equations of some interesting
statistical observables of the very general jump processes with multiple
internal states. Finally, we consider the associated initial and boundary value
problems and prove their well-posedness of the Galerkin weak formulation in
$\mathbb{R}^n$. To obtain the coercivity, we claim that the probability density
function $m(Y)$ should be nondegenerate.
