Non-reversible processes: GENERIC, hypocoercivity and fluctuations

We consider two approaches to study non-reversible Markov processes, namely the hypocoercivity theory and 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 characterise 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.