  Let $Q$ be a differential operator of order $\leq 1$ on a complex metric
vector bundle $\mathscr{E}\to \mathscr{M}$ with metric connection $\nabla$ over
a possibly noncompact Riemannian manifold $\mathscr{M}$. Under very mild
regularity assumptions on $Q$ that guarantee that $\nabla^{\dagger}\nabla/2+Q$
generates a holomorphic semigroup $\mathrm{e}^{-zH^{\nabla}_{Q}}$ in
$\Gamma_{L^2}(\mathscr{M},\mathscr{E})$ (where $z$ runs through a complex
sector which contains $[0,\infty)$), we prove an explicit Feynman-Kac type
formula for $\mathrm{e}^{-tH^{\nabla}_{Q}}$, $t>0$, generalizing the standard
self-adjoint theory where $Q$ is a self-adjoint zeroth order operator. For
compact $\mathscr{M}$'s we combine this formula with Berezin integration to
derive a Feynman-Kac type formula for an operator trace of the form $$
\mathrm{Tr}\left(\widetilde{V}\int^t_0\mathrm{e}^{-sH^{\nabla}_{V}}P\mathrm{e}^{-(t-s)H^{\nabla}_{V}}\mathrm{d}
s\right), $$ where $V,\widetilde{V}$ are of zeroth order and $P$ is of order
$\leq 1$. These formulae are then used to obtain a probabilistic
representations of the lower order terms of the equivariant Chern character (a
differential graded extension of the JLO-cocycle) of a compact even-dimensional
Riemannian spin manifold, which in combination with cyclic homology play a
crucial role in the context of the Duistermaat-Heckmann localization formula on
the loop space of such a manifold.
