  We consider two approaches to study non-reversible Markov processes, namely
the Hypocoercivity Theory (HT) and GENERIC (General Equations for
Non-Equilibrium Reversible-Irreversible Coupling); the basic idea behind both
of them is to split the process into a reversible component and a
non-reversible one, and then quantify the way in which they interact. We
compare such theories and provide explicit formulas to pass from one
formulation to the other; as a bi-product we give a simple proof of the link
between reversibility of the dynamics and gradient flow structure of the
associated Fokker-Planck equation. We do this both for linear Markov processes
and for a class of nonlinear Markov process as well. We then characterize the
structure of the Large deviation functional of generalised-reversible
processes; this is a class of non-reversible processes of large relevance in
applications. Finally, we show how our results apply to two classes of Markov
processes, namely non-reversible diffusion processes and a class of Piecewise
Deterministic Markov Processes (PDMPs), which have recently attracted the
attention of the statistical sampling community. In particular, for the PDMPs
we consider we prove entropy decay.
