  We consider a complete noncompact smooth Riemannian manifold $M$ with a
weighted measure and the associated drifting Laplacian. We demonstrate that
whenever the $q$-Bakry-\'Emery Ricci tensor on $M$ is bounded below, then we
can obtain an upper bound estimate for the heat kernel of the drifting
Laplacian from the upper bound estimates of the heat kernels of the Laplacians
on a family of related warped product spaces. We apply these results to study
the essential spectrum of the drifting Laplacian on $M$.
