  Let $\mathscr{L}$ be either the Laplace--Beltrami operator, its shift without
spectral gap, or the distinguished Laplacian on a symmetric space of noncompact
type $\mathbb{X}$ of arbitrary rank. We consider the heat equation, the
fractional heat equation, and the Caffarelli--Silvestre extension problem
associated with $\mathscr{L}$, and in each of these cases we characterize the
weights $v$ on $\mathbb{X}$ for which the solution converges pointwise a.e. to
the initial data when the latter is in $L^{p}(v)$, $1\leq p < \infty$. As a
tool, we also establish vector-valued weak type $(1,1)$ and $L^{p}$ estimates
($1<p<\infty$) for the local Hardy--Littlewood maximal function on
$\mathbb{X}$.
