The existence of the heat kernel of the Laplace-Beltrami operator on a compact Riemannian manifold is well known (e.g. through the construction via a short time expansion coupled with a Volterra series in "Heat Kernels and Dirac Operators" by Berline, Getzler, and Vergne).
In the setting of non-compact Riemannian manifolds, I have heard numerous times that the above construction, together with an exhaustion of the compact manifold by compact sets, yields the existence of heat kernel by use of the maximum principle. I am requesting a reference to some notes or a book where this is discussed at an accessible level (preferably without pseudo-differential calculus).
Thank you!