Heat kernel estimates and parabolic Harnack inequalities on graphs and resistance forms
This paper is an invited contribution to the special issue of Publications of RIMS. It presents recent work (partly published at that time only in preprint form) of the author and also joint work of the author with R. F. Bass, M. T. Barlow, T. Coulhon. The framework for the main result is that of a measure metric space, on which a resistance form, as defined by \textit{J. Kigami} [``Analysis on fractals'' (2001; Zbl 0998.28004)] is given. With this resistance form, one associates a heat kernel. The main result asserts the equivalence between a form of upper and lower estimate for the heat kernel, and a uniform volume growth with volume doubling condition. Examples (on trees and fractals), where such upper or lower estimates hold are presented in a final chapter.
scientific article; zbMATH DE number 2151103