In "Heat Kernels and Dirac Operators" by Berline, Getzler and Vergne, the authors construct a (unique) heat kernel associated to any generalized Laplacian in a vector bundle over a Riemannian manifold.
Could you please point me to texts doing the same kind of construction (not necessarily using densities) in vector bundles over arbitrary (i.e. not necessarily compact anymore) Riemannian manifolds for the connection Laplacian?
I am mostly interested in the clear definition of the object and the statement of its basic properties. Do the positivity and minimality properties of the heat kernel on functions have analogous properties in bundles?