We characterize Gaussian estimates for transition probability of a discrete time Markov chain in terms of geometric properties of the underlying state space. In particular, we show that the following are equivalent: (1) Two sided Gaussian bounds on heat kernel (2) A scale invariant Parabolic Harnack inequality (3) Volume doubling property and a scale invariant Poincaré inequality. The underlying state space is a metric measure space, a setting that includes both manifolds and graphs as special cases. An important feature of our work is that our techniques are robust to small perturbations of the underlying space and the Markov kernel. In particular, we show the stability of the above properties under quasi-isometries. We discuss various applications and examples.
Both the authors were partially supported by NSF grants DMS 1004771 and DMS 1404435. The first author was supported in part by NSERC (Canada) and the Canada research chairs program.
"Harnack inequalities and Gaussian estimates for random walks on metric measure spaces." Electron. J. Probab. 28 1 - 81, 2023. https://doi.org/10.1214/23-EJP954