Non-reversible processes: GENERIC, Hypocoercivity and fluctuations
share
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.
READ FULL TEXT