Characterizing near-bipartiteness, Harnack inequalities, and heat kernel estimates

This thesis consists of three projects. In the first project, we show that if the mixing time of a reversible, aperiodic, irreducible Markov chain is much larger than the lazy mixing time, then the Markov chain is almost bipartite, in a precise quantitative sense that we describe. Along the way, we prove that the gap between the second smallest eigenvalue of the transition matrix and −1 is at least a universal constant times the gap between the second largest eigenvalue and 1. In the second project, we develop probabilistic methods which in a broad class of cases allow us to determine whether an isotropic unimodal Lévy jump process on Euclidean space satisfies the elliptic Harnack inequality (EHI), by looking only at the jump kernel and its truncated second moments. Both our positive results and our negative results can be applied to subordinated Brownian motions (SBMs) in particular. We prove EHI for many jump processes which previous methods were unable to determine it for, and also produce the first known example of an SBM that does not satisfy EHI. In the third project, we show that a result of Chen, Kumagai, and Wang—on the stability of heat kernel estimates and parabolic Harnack inequality for symmetric non-local Dirichlet forms—can be extended to certain graphs.