Relaxation times of non-reversible Markov processes

Abstract

We develop a systematic approach to quantify $L^2$-relaxation times for non-reversible Markov processes based on the singular value gap of the generator introduced by Chatterjee. The inverse of the singular value gap is equivalent to the relaxation time of the time-averaged transition semigroup. We show that, moreover, the inverse singular value gap of the two-point motion is equivalent to the relaxation time of the non-averaged transition semigroup, and that it provides a lower bound on the usual spectral gap of the generator. We then introduce a method for deriving lower bounds on singular value gaps for Markov processes with degenerate noise that is based on the concept of a first- and second-order collapse of the generator. It follows ideas from hypocoercivity developed in a previous series of works but is simpler and more broadly applicable. In contrast to previous results, it includes settings with non-vanishing first-order collapse, and thus applies directly to Markov chains (in continuous time), but also to diffusion processes and piecewise-deterministic Markov processes. Our approach yields sharp upper and lower bounds for several classes of examples. First applications include the proof of a conjecture by Diaconis and Miclo on a square-root speed-up for lifted random walks on abelian groups, as well as bounds on relaxation times of switching flows perturbed by noise and of non-reversible diffusion processes.

arXiv commentStrengthened relation between non-averaged relaxation time and singular value gap of the two-point motion, eliminating the need for regular initial laws. 45 pages