  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.
