Peskun-Tierney ordering for Markov chain and process Monte Carlo: beyond the reversible scenario

Historically time-reversibility of the transitions or processes underpinning Markov chain Monte Carlo methods (MCMC) has played a key r\^ole in their development, while the self-adjointness of associated operators together with the use of classical functional analysis techniques on Hilbert spaces have led to powerful and practically successful tools to characterize and compare their performance. Similar results for algorithms relying on nonreversible Markov processes are scarce. We show that for a type of nonreversible Monte Carlo Markov chains and processes, of current or renewed interest in the Physics and Statistical literatures, it is possible to develop comparison results which closely mirror those available in the reversible scenario. We show that these results shed light on earlier literature, proving some conjectures and strengthening some earlier results.