Gaussian upper heat kernel bounds on Riemannian manifolds with a doubling measure are characterized by Sobolev inequalities in arbitrarlily small balls. Since the small-time behaviours of heat kernels of manifolds and graphs differ significantly, their Gaussian behaviour at time zero must be different as well. In particular, the Gaussian behaviour of the heat kernel of a graph cannot be equivalent to Sobolev inequalities in arbitrarily small balls. Given a graph whose measure is doubling only on large balls, I will present a new characterization of Gaussian upper bounds for large times in terms of a new variant of Sobolev inequalities valid on large balls. This is an ongoing and in parts joint work with Matthias Keller.
Contact: Marcel Schmidt