Harnack inequalities and Gaussian estimates for random walks on metric measure spaces

TL;DR

This paper establishes the equivalence of Gaussian bounds, Harnack inequalities, and geometric properties like volume doubling and Poincaré inequalities for Markov chains on metric measure spaces, with robustness under perturbations.

Contribution

It characterizes Gaussian estimates for Markov chains in terms of geometric properties and proves their stability under quasi-isometries in metric measure spaces.

Findings

Gaussian bounds are equivalent to Harnack inequalities and volume doubling.

Results are stable under small perturbations and quasi-isometries.

Applicable to manifolds, graphs, and other metric measure spaces.

Abstract

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 Poincar\'{e} 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.