  Let $M$ be a complete Riemannian manifold and $D\subset M$ a smoothly bounded
domain with compact closure. We use Brownian motion and the classic results on
the Stieltjes moment problem to study the relationship between the Dirichlet
spectrum of $D$ and the heat content asymptotics of $D.$ Central to our
investigation is a sequence of invariants associated to $D$ defined using exit
time moments. We prove that our invariants determine that part of the spectrum
corresponding to eigenspaces which are not orthogonal to constant functions,
that our invariants determine the heat content asymptotics associated to the
manifold, and that when the manifold is a generic domain in Euclidean space,
the invariants determine the Dirichlet spectrum.
