Let \(p(x, y, t)\) be a heat kernel of a noncompact Riemannian manifold. The paper surveys the recent authors’ results on large time behaviour of the heat kernel with some extensions, sharpenings and examples provided. The main topic are lower and upper bounds for the kernel of the form \(t^{-a/2} \exp (-d^2 (x,y)/ ct)\), where \(d(x,y)\) is the Riemannian distance on the manifold. An example of the results discussed in the paper: Let \(M\) be a complete noncompact Riemannian manifold with Ricci curvature bounded from below and such that the volume of the metric disk \(V(R)\) of the radius \(R\) is of the order of \(R^a\). Moreover, assume that the fundamental Neumann eigenvalue for \(V(R)\) is bounded from below by \(\text{const}\cdot R^{-2}\) for large \(R\) and for large \(t\) the upper bound of the form given above holds for the kernel. Then the lower bound for the kernel is of the same form.
For the entire collection see [Zbl 0814.00017].