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Historically time-reversibility of the transitions or processes underpinningMarkov chain Monte Carlo methods (MCMC) has played a key r\^ole in theirdevelopment, while the self-adjointness of associated operators together withthe use of classical functional analysis techniques on Hilbert spaces have ledto powerful and practically successful tools to characterize and compare theirperformance. Similar results for algorithms relying on nonreversible Markovprocesses are scarce. We show that for a type of nonreversible Monte CarloMarkov chains and processes, of current or renewed interest in the Physics andStatistical literatures, it is possible to develop comparison results whichclosely mirror those available in the reversible scenario. We show that theseresults shed light on earlier literature, proving some conjectures andstrengthening some earlier results.

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