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 know in the setting of diffusion over Riemannian manifolds (or more generally local Dirichlet spaces) and random walks over graphs. However this random walk over a continuous metric measure space raises new difficulties. I will explain some of these difficulties and how to overcome them. We will discuss some examples and applications.

This is joint work with Laurent Saloff-Coste. (work in progress)