Relaxation times of non-reversible Markov processes

2026-07-12

AI Summary

This work addresses the quantification of L² relaxation times for irreversible Markov processes by establishing a unified analytical framework based on the singular value gap of the generator and its hypoelliptic structure. The approach applies seamlessly to continuous-time Markov chains, diffusions, and piecewise deterministic processes, encompassing both first- and second-order hypoelliptic settings under degenerate noise. It extends and simplifies the theory of hypocoercivity, yielding sharp upper and lower bounds on relaxation times. As notable applications, the method confirms the conjecture of Diaconis and Miclo that lifted random walks on Abelian groups achieve a square-root speedup, and provides precise convergence rate estimates for noisy switching flows and irreversible diffusions.

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 singular value gap of the two-point motion also provides a lower bound on the usual spectral gap of the generator, and its inverse provides upper bounds on relaxation times without time averaging for sufficiently regular initial laws. 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.

Problem

• non-reversible Markov processes
• relaxation times
• singular value gap
• spectral gap
• hypocoercivity

Innovation

• singular value gap
• non-reversible Markov processes
• hypocoercivity
• relaxation time
• two-point motion