  In this article, we examine the general space-time fractional diffusion
equation for left-invariant hypoelliptic homogeneous operators on graded Lie
groups. Our study covers important examples such as the time-fractional
diffusion equation, the space-time fractional diffusion equation when diffusion
is under the influence of sub-Laplacian on the Heisenberg group, or general
stratified Lie groups. We establish the global well-posedness of the Cauchy
problem for the general space-time fractional diffusion equation for the
Rockland operator on a graded Lie group in the associated Sobolev spaces. More
precisely, we establish the existence and uniqueness results for both
homogeneous and inhomogeneous fractional diffusion equations. In addition, we
also develop some regularity estimates.
