Christian Rose (Universität Bremen)
Abstract: The heat kernel as the minimal fundamental solution of the heat equation is one of the most important objects studied in geometric analysis and encodes the geometry of the underlying space in analytic terms. On graphs it is nowadays known that the short time behaviour of the heat kernel is very different from the short time behaviour of heat kernels on manifolds due to the non-locality of the graph Laplacian. However, on large scales, i.e., large times or distances, heat kernels on graphs satisfying certain regularity properties are expected to behave like heat kernels on manifolds with corresponding properties. Our main result presented in this talk will be an optimal Gaußian decay estimate for large times and distances for any graph satisfying volume doubling and Sobolev inequalities on large balls. The explicit estimates depend on $L^p$-means of the vertex degree and inverted vertex measure for suitable $p$ depending on the Sobolev and doubling dimension and hold, e.g., on antitrees with slowly increasing sphere functions.
This is a joint work with Matthis Keller.
