In this talk, I will describe some recent results on the structure of ancient solutions of the heat equation on certain Riemannian manifolds (by Colding and Minicozzi, and Zhang). I will describe some versions of these results in the more general setting of Dirichlet spaces, which further include for example many fractal spaces and infinite dimensional spaces. In particular, I will show how to adapt their methods in the situation where there are only cutoff functions with “bounded energy”, and how to show that the time derivatives of an ancient weak solution of the heat equation is still an ancient weak solution.

This is joint work with Laurent Saloff-Coste.