Random walks on metric measure spaces

A metric space is a length space if the distance between two points equals the infimum of the lengths of curves joining them. For a measured length space, we characterize Gaussian estimates for iterated transition kernel for random walks and parabolic Harnack inequality for solutions of a corresponding discrete time version of heat equation by geometric assumptions (Poincaré inequality and Volume doubling property). Such a characterization is well known in the setting of diffusion over Riemannian manifolds (or more generally local Dirichlet spaces) and random walks over graphs (due to the works of A. Grigor'yan, L. Saloff-Coste, K. T. Sturm, T. Delmotte, E. Fabes & D. Stroock). However this random walk over a continuous space raises new difficulties. I will explain some of these difficulties and how to overcome them. We will discuss some motivating examples and applications.
This is joint work with Laurent Saloff-Coste. (in preparation)