  Heat kernels arise in a variety of contexts including probability, geometry,
and functional analysis; the automorphic heat kernel is particularly important
in number theory and string theory. The typical construction of an automorphic
heat kernel as a Poincar\'{e} series presents analytic difficulties, which can
be dealt with in special cases (e.g. hyperbolic spaces) but are often
sidestepped in higher rank by restricting to the compact quotient case. In this
paper, we present a new approach, using global automorphic Sobolev theory, a
robust framework for solving automorphic PDEs that does not require any
simplifying assumptions about the rank of the symmetric space or the
compactness of the arithmetic quotient. We construct an automorphic heat kernel
via its automorphic spectral expansion in terms of cusp forms, Eisenstein
series, and residues of Eisenstein series. We then prove uniqueness of the
automorphic heat kernel as an application of operator semigroup theory.
Finally, we prove the smoothness of the automorphic heat kernel by proving that
its automorphic spectral expansion converges in the $C^\infty$-topology.
