  Incomplete cusp edges model the behavior of the Weil-Petersson metric on the
compactified Riemann moduli space near the interior of a divisor. Assuming such
a space is Witt, we construct a fundamental solution to the heat equation, and
using a precise description of its asymptotic behavior at the singular set, we
prove that the Hodge-Laplacian on differential forms is essentially
self-adjoint, with discrete spectrum satisfying Weyl asymptotics. We go on to
prove bounds on the growth of $L^2$-harmonic forms at the singular set and to
prove a Hodge theorem, namely that the space of $L^2$-harmonic forms is
naturally isomorphic to the middle-perversity intersection cohomology.
Moreover, we develop an asymptotic expansion for the heat trace near $t = 0$.
