  We consider weak solutions of the fractional heat equation posed in the whole
$n$-dimensional space, and establish their asymptotic convergence to the
fundamental solution as $t\to\infty$ under the assumption that the initial
datum is an integrable function, or a finite Radon measure. Convergence with
suitable rates is obtained for solutions with a finite first initial moment,
while for solutions with compactly supported initial data convergence in
relative error holds. The results are applied to the fractional Fokker-Planck
equation. Brief mention of other techniques and related equations is made.
